Symbol reference

Language-reference pages for compute-engine functions, generated from verified example data. Each example is re-evaluated live and pinned by a test, so the docs can't drift from the engine.

Elsewhere

Every symbol page opens with where the same thing lives in other systems — Wikipedia, MathWorld, Wikidata, DLMF, Fungrim, the OEIS, FindStat, the Wolfram Language, SageMath, SymPy, mpmath, mathlib4 and more. Nothing there is typed twice: a chip's tooltip says which of the crosswalk's sources it came from.

written on the entry, or on one call form when the head means something else at another arity (Zeta(s, a) is Hurwitz's function and links to Hurwitz's pages)
curated in the reference package's hand-kept table
the enumeratio catalog's own crosswalk, which is where FindStat ids and Sage classes come from; a carrier domain inherits what its collections recorded
compute-engine's own Wikidata id — audited, since 41 of the 101 the engine declares are dead or about something else (PlanckConstant pointed at Mount Vesuvius) — and everything that item points at
the DLMF's defining equation, found in the handbook's index of notations by the names the head already has
FindStat, asked by value: every object of the carrier up to a size, our definition evaluated on each, and the finder's answer kept only where it agrees on everything
the OEIS, asked by count: a family's own count kernel for the first sizes, searched, and kept where the terms agree once aligned

Four of those are checks rather than pointers, and the chips say so. A ✓ means the claim was put to a computation — a definition evaluated against theirs, a count against their terms, this head's own examples run in that kernel, or a Fungrim identity instantiated and evaluated on both sides — and the tooltip says how many. A ! means the same computation ran and something disagreed, which is the more interesting outcome: every one of them so far has been a bug on our side, and the entry says which.

The scripts in @enumeratio/reference keep it honest: crosswalk:collect (offline, test-guarded), crosswalk:fetch (the outside indexes), crosswalk:find (FindStat by value, the OEIS by count), crosswalk:verify (Fungrim's identities, numerically), crosswalk:audit (the engine's Wikidata ids) and crosswalk:check (follows every link).

Fungrim: the symbol page, and every identity the engine compiled that mentions the head
the Wolfram symbol the transpiler vouches for, and the oracle's equivalent call in each kernel — resolved to its documentation anchor through the system's Sphinx inventory

Prose can lean on the same table: <Symbol type="sage">SetComposition</Symbol> is our name, and the link is whatever Sage calls it.

2307 heads — 14 defined by a reference expression, 61 primitive on the declared frontier, and 2232 not yet placed either way.

Elsewhere: 8015 pointers from 2408 of 3070 pages into 22 systems — Fungrim entry 2623 ✓1281, FindStat 1287 ✓1092, Wolfram Language 818 ✓627, OEIS 539 ✓109, SageMath 502 ✓330, Wikipedia 326, Wikidata 264, MathWorld 256, SymPy 210, Fungrim 186 ✓97, Julia 177, mpmath 163, mathlib4 112, DLMF 104, Rust 96, Britannica 91, nLab 75, Encyclopedia of Mathematics 71, Oscar 44, Combinatorica 39 ✓39, Rosetta Code 21, MATLAB 11.

Adèles and idèles

Adele -- An adèle: a real number $r$ beside a profinite number $z \in \hat{\mathbb{Q}}$ . Adele(q) for rational $q$ is the principal adèle, $q$ at every place.
Idele -- An idèle: a unit of the adèle ring — a real $r$ together with $s \in \mathbb{Q}^\times$ principal everywhere except at a listed finite set of primes, where distinguished units override it.
ProfiniteDecomposition -- $\{b, a\}$ with $m = b \cdot a$ , $b \in GL_n(\hat{\mathbb{Z}})$ and $a \in GL_n^+(\mathbb{Q})$ upper triangular — strong approximation for a matrix over $\hat{\mathbb{Q}}$ .
ProfiniteNumber -- A profinite number: the rational $x + m\hat{\mathbb{Z}}$ in $\hat{\mathbb{Q}} = \hat{\mathbb{Z}} \otimes \mathbb{Q}$ , known modulo $m$ . Modulus 0 (the default) is just the rational $x$ itself.
ProfinitePlot -- The graph of $f: \hat{\mathbb{Z}} \to \hat{\mathbb{Z}}$ in the variable $x$ , at precision $k$ (default 5) — Lenstra's picture of a profinite function, drawn as an ArrayPlot.

Alternating sign matrix statistics

AlternatingSignMatrixPosetRank -- Half the sum of (i - j)^2 a(i, j): the rank in the alternating sign matrix poset.
DecreasableCornerSumPositions -- The interior positions of the corner sum matrix whose entry can be lowered by one to give another alternating sign matrix's corner sum matrix.
DistinctDiagonalSumCount -- The number of distinct sums of the entries along a diagonal.
FirstColumnOnePosition -- The row of the unique 1 in the first column.
FirstRowOnePosition -- The column of the unique 1 in the first row.
LastColumnOnePosition -- The row of the unique 1 in the last column.
LastRowOnePosition -- The column of the unique 1 in the last row.
MainDiagonalSum -- The sum of the entries on the main diagonal, its trace.
MatrixInversionNumber -- The sum of a(i, j) a(k, l) over the pairs of entries with i > k and j < l.
MatrixPermanent -- The permanent: the sum over the permutations s of the product of a(i, s(i)).
MaximalDiagonalNonzeroCount -- The largest number of nonzero entries along a diagonal.
MaximalDiagonalSum -- The largest sum of the entries along a diagonal.
MinimalRowZeroCount -- The fewest zeros in a row.
NegativeOneCount -- The number of entries equal to -1.
NonzeroEntryCount -- The number of nonzero entries.
OnesOnMainDiagonal -- The number of 1s on the main diagonal.
PositiveMatrixInversionNumber -- The inversion number less the number of entries equal to -1.
PositiveOneCount -- The number of entries equal to 1.
TopLeftZeroSquareSize -- The side of the largest square of zeros at the top left corner.
ZerosOnMainDiagonal -- The number of 0s on the main diagonal.

Analytic 2 primitive

Assuming -- Evaluate $expr$ with $cond$ assumed for the duration of the call, then forget it. Provided by @enumeratio/analytic.
CaputoDkernel -- The Caputo fractional derivative of order $\alpha > 0$ of $f$ with respect to $x$ , at base point 0.
DSolveValuekernel -- Solves a linear, constant-coefficient ODE of order 1 or 2 for Y(x), with optional initial conditions.
Piecewise -- A conditional value: List of {value, condition} pairs, tried in order, plus a $default$ (0 when omitted) for when none holds. Provided by @enumeratio/analytic.
PiecewiseExpand -- Rewrite Abs, Sign, Min, Max, UnitStep, Clip, UnitBox, UnitTriangle and, over a bounded variable, Floor, Ceil, Round, IntegerPart, FractionalPart, Mod, Quotient and the waves inside $expr$ into Piecewise, recursively. Provided by @enumeratio/analytic.
Refine -- Simplify $expr$ using $cond$ (or, with no $cond$ , whatever is already assumed) as a fact — the same fact-driven simplification Assuming scopes for a whole subcomputation, but applied once. Provided by @enumeratio/analytic.
SeriesCoefficient -- The coefficient of $(x-x_0)^n$ in the Taylor series of $f$ about $x_0$ , via $\frac{f^{(n)}(x_0)}{n!}$ . Provided by @enumeratio/analytic.

Arithmetic

Abs -- The absolute value or magnitude of x: $|x|$ , its distance from 0.
Ceil -- The least integer greater than or equal to x: $\lceil x \rceil$ .
Chop -- Replaces a real number very close to 0 with exact 0.
Clamp -- Constrains x to the range [lower, upper].
Floor -- The greatest integer less than or equal to x: $\lfloor x \rfloor$ .
IsEven -- Whether n is an even integer.
IsOdd -- Whether n is an odd integer.
Max -- The largest of its arguments.
Min -- The smallest of its arguments.
N -- A numeric value of x -- to d significant digits, every one of them right.
Negate -- The additive inverse of x: $-x$ .
Rationalize -- Finds a simple rational number near x.
Root -- The nth root of x: $\sqrt[n]{x} = x^{1/n}$ .
Round -- Rounds x to the nearest integer, or to the nearest multiple of a.
Sign -- The sign of x: -1, 0, or 1 for negative, zero, or positive x.
Sqrt -- The principal square root of x: $\sqrt{x}$ .
Square -- The second power of x: $x^2$ .
Truncate -- The integer part of x, rounding toward 0.

Binary word statistics

Balance -- The least q such that any two factors of one length differ by at most q ones.
BalancedPrefixCount -- The nonempty prefixes with as many 0s as 1s.
BaselineCrossings -- The times the path, a 1 a step up and a 0 a step down, crosses the baseline.
BorderCount -- The prefixes shorter than the word that are also suffixes, the empty one included.
BorderLength -- The length of the longest proper prefix that is also a suffix; 0 for none.
CatalanFactorCount -- The factors in the Catalan factorization: the critical steps, and the middle Dyck word if it is not empty.
ChangeCount -- The positions i where the letters i and i + 1 differ.
CriticalStepCount -- The critical steps l + m in the Catalan factorization (D 0)^l D (1 D)^m.
CyclicShiftIndex -- The position of the word among its distinct cyclic shifts in lexicographic order, from 0.
DecimalRepresentation -- The word read as a binary numeral.
DecimalRepresentationWithLeadingOne -- The word with a 1 put in front, read as a binary numeral.
DescentWordPreimageCount -- The permutations whose descent word is this word.
DistinctCubeCount -- The distinct nonempty words u with uuu a factor.
DistinctFactorCount -- The distinct factors, the empty one included: the subword complexity.
DistinctSquareCount -- The distinct nonempty words u with uu a factor.
DistinctSubsequenceCount -- The distinct subsequences, the empty one included.
EvenPalindromicPrefixHalfLength -- Half the length of the longest palindromic prefix of even length.
FirstOnePositionAfterAppendedOne -- The position of the first 1 once a 1 is appended to the word: one more than the leading zeros.
Flex -- The product of the word's lex (its position among its distinct cyclic shifts, from 1) and its frequency.
HammingWeight -- The number of ones.
InternalInversions -- The pairs i < j with w(i) > w(j) that lie in the same block of the sorted word.
LatticePathsAboveWordCount -- The paths of the word's length, ending anywhere, that stay weakly above the path the word gives.
LeadingOnesCount -- The number of ones before the first 0.
LongestAlternatingFactor -- The length of the longest factor in which every two neighboring letters differ.
LongestCatalanFactorSemilength -- The semilength of the longest Dyck word in the Catalan factorization.
LongestPalindromicFactor -- The length of the longest factor that reads the same backwards.
LongestPalindromicFactorStartingWithOne -- The length of the longest palindromic factor that begins with a 1.
LongestPalindromicPrefix -- The length of the longest prefix that reads the same backwards.
LongestPalindromicSubsequence -- The length of the longest subsequence that reads the same backwards.
LongestRunOfOnes -- The length of the longest run of ones.
LongestYamanouchiPrefix -- The length of the longest prefix in which no prefix has more ones than zeros.
LyndonFactorizationLength -- The factors in the Lyndon factorization.
MinimalPeriod -- The least p with w(i) = w(i + p) wherever both are letters.
OddOneRunCount -- The number of maximal runs of ones of odd length.
OddPalindromicPrefixHalfLength -- The largest j such that the prefix of length 2 j + 1 is a palindrome.
OnePositionSum -- The sum of the positions of the ones.
OneRunCount -- The number of maximal runs of ones.
OnesMinusZeros -- The number of ones less the number of zeros.
PalindromicDecompositionCount -- The ways to write the word as a concatenation of palindromic factors.
PathDepth -- The most zeros less ones among the prefixes.
PrimitiveFactorCount -- The distinct nonempty factors that are not a power of a shorter word.
RepetitionExponent -- The largest e such that the word is e copies of one word.
ReverseComplementFactorHalfLength -- Half the length of the longest factor that is its own reverse complement.
ReverseComplementFactorHalfLengthStartingWithOne -- Half the length of the longest factor that is its own reverse complement and begins with a 1.
SameEndpointPathsAboveWordCount -- The paths with the word's number of ones and zeros that stay weakly above the path the word gives.
ShortestPalindromicDecompositionLength -- The fewest palindromic factors the word is a concatenation of.
StrictlyIncreasingRuns -- The maximal runs of strictly increasing letters.
SymmetricBorderLength -- The length of the longest prefix whose reverse is a suffix.
UniqueFactorLength -- The least length for which all factors of that length are different.
ValleyFillingDegree -- The number of times every valley 01 must be swapped for a peak 10, and a trailing 0 made a 1, to leave no zeros.

Boxes

AnimatorBox -- A slider that plays: Animator lowers to it.
ArrowBox -- A polyline with an arrowhead at its end: Points.
ButtonBox -- A link: BaseStyle -> "Hyperlink" to a URL, BaseStyle -> "Link" to a head's page.
CheckboxBox -- A box that is ticked or not: Checkbox lowers to it.
ColorSetterBox -- A swatch or a spectrum to pick a color from: ColorSlider lowers to it.
CuboidBox -- An axis-aligned box in space, from Min to Max.
DiskBox -- A disk, or a sector of one: Radius, a Center unless its address gives one, and Angles for a sector.
DisplayForm -- Boxes shown as what they draw, rather than as the expression that describes them.
DynamicBox -- An expression the environment re-evaluates when the variables it reads move: what Dynamic lowers to.
DynamicModuleBox -- A scope: the controls and readouts inside bind the same variables. DynamicModule lowers to it.
ErrorBox -- Boxes marked as an error.
FormBox -- Boxes in a form: TeXForm holds TeX as written, StandardForm or TraditionalForm formula boxes.
FractionBox -- A numerator over a denominator.
FrameBox -- A box drawn with a frame around it.
Graphics3DBox -- A drawing in space: 3-D primitives in a frame's coordinates, looked at from a ViewPoint.
GraphicsBox -- A drawing: primitives in a frame's coordinates, with ColorRules, ColorMixing, BoundaryStyle, Selection and ViewKind as options.
GraphicsComplex3DBox -- 3-D primitives that share vertices: the first operand lists them, and a primitive names its own by Indices.
GraphicsComplexBox -- Primitives that share places: the coordinates give each address's position (Addresses names them), and a primitive tagged with an address sits there unless it names its own.
GridBox -- A two-dimensional array of boxes: a matrix's body, a piecewise definition's cases.
InputFieldBox -- A field you type a value into: InputField lowers to it.
InsetBox -- A box placed in a drawing: text at its Center, Size high, at Opacity.
InterpretationBox -- Boxes that display one way and stand for exactly the expression they carry.
IntervalSliderBox -- A track with two thumbs that bound an interval: IntervalSlider lowers to it.
KnobBox -- A number you drag in place, with a ladder of step sizes: Knob lowers to it.
Line3DBox -- A path in space through Points (or a complex's Indices), VertexColors giving the color at each point.
LineBox -- A polyline: Points, and Breaks, the indices where the pen lifts.
ListPickerBox -- A list of entries to pick from: ListPicker lowers to it.
LocatorBox -- A point handle you drag inside a figure: Locator lowers to it.
MakeBoxes -- The boxes of an expression's traditional notation, without evaluating it.
OverscriptBox -- A box set above a base: $\overline{3}$ , $\hat{x}$ .
PaneBox -- A box at a size, scrolling or scaling past it: ImageSize fixes the room, Scrollbars and ImageSizeAction say what happens to the overflow.
PaneSelectorBox -- Alternatives, one shown: the pane whose key is the binding's value.
PanelBox -- A box on a background with a frame, padded: what Panel lowers to.
Point3DBox -- A set of points in space: Points.
PointBox -- A set of points: Points.
Polygon3DBox -- Polygons in space: Points for one, or Indices into a complex's points for many, each filled by its entry of FaceColors.
PolygonBox -- A filled polygon: its Points.
PolyhedronBox -- A solid drawn as its boundary: the rings of its Faces.
PopupMenuBox -- A menu of entries, one chosen: PopupMenu lowers to it.
RadicalBox -- An $n$ th-root sign over a box, the index in its crook.
RadioButtonBox -- One radio button, on when the variable holds its entry; a RadioButtonBar is a row of them.
RasterBox -- A pixel grid in a drawing's coordinates: one color per cell in Colors, spread over DataRange.
RawBoxes -- Boxes to be drawn in place, inside a larger expression.
RectangleBox -- An axis-aligned rectangle: its corners Min and Max.
RowBox -- A row of boxes, juxtaposed left to right: the box a sequence of tokens makes.
RowSource -- The rows of an ordered collection, by range: a count that is exact, a growing lower bound or infinite, and each row as boxes.
SetterBox -- One button, pressed when the variable holds its entry; a SetterBar is a row of them.
Slider2DBox -- A square with a draggable point that moves a pair: Slider2D lowers to it.
SliderBox -- A track with a thumb that moves a number: Slider lowers to it.
SqrtBox -- A square-root sign over a box.
StepperBox -- An integer in a sentence, stepped in place with arrows.
StyleBox -- A box drawn in a style: FontColor, FontWeight, FontSlant, FontSize, Background.
SubscriptBox -- A base with a subscript: $x_1$ .
SubsuperscriptBox -- A base with a subscript and a superscript: $\int_0^1$ .
SuperscriptBox -- A base with a superscript: $x^2$ .
TableViewBox -- A scrolling grid over a row source: it asks for the rows in view, so a massive or infinite collection draws without ever being held.
TagBox -- Boxes labelled with a tag saying how to read them back as an expression.
TemplateExpression -- A hole in prose holding an expression's Epsil, filled by the environment it renders in.
TemplateSlot -- A named hole in prose, filled by the environment it renders in.
TextBox -- Literal text inside boxes, set as text rather than read as tokens.
TextCell -- A block of prose in a cell style: Text, Section, Item, DisplayFormula, Program, …
TextData -- Running text; its string leaves are words, not tokens.
ToBoxes -- The boxes an expression's traditional notation is made of, after evaluating it.
TogglerBox -- A button that cycles through its entries: Toggler lowers to it.
TooltipBox -- A box and what shows on hover or focus: what Tooltip lowers to.
UnderoverscriptBox -- A base with boxes below and above it: $\sum_{n=1}^{10}$ .
UnderscriptBox -- A box set below a base: the limit under $\lim$ .

Braids and knots

AlexanderPolynomial -- The Alexander polynomial of a knot, computed from the reduced Burau representation over $\mathbb{Z}[t,t^{-1}]$ and returned as an ordinary expression in $t$ . Takes the knot however it is named — as $T(p,q)$ , as a braid it closes from, or as a modular word.
BracketInvariant -- The writhe-corrected Kauffman bracket $(-A^3)^{-w}\langle L\rangle$ of a knot or link — already an invariant, unlike KauffmanBracket on its own.
Braid -- A braid in Artin's presentation: $n$ strands and a word whose letter $k$ means $\sigma_k$ and $-k$ means $\sigma_k^{-1}$ . Closing it up names a link — and by Alexander's theorem, every link.
BraidClosure -- The link a braid closes to — each out strand joined to the in slot below it, round the side. By Alexander's theorem every link is one, and a cell that evaluates it draws the closed braid.
BraidComponents -- The number of components in a braid's closure — the cycle count of its BraidPermutation.
BraidCrossings -- The number of crossings in a braid word — its length.
BraidInverse -- The inverse of a braid — its word reversed with every letter negated.
BraidIsKnot -- Whether a braid's closure is a knot rather than a multi-component link — its underlying permutation is a single cycle.
BraidIsPositive -- Whether every letter in a braid's word is positive — no generator appears inverted.
BraidPermutation -- The image of a braid in the symmetric group — forget which strand went over, and keep only where each strand ends up.
BraidPower -- A braid raised to an integer power — its word repeated $k$ times, or the inverse's word repeated $|k|$ times when $k$ is negative.
BraidProduct -- The product of two braids on the same number of strands — one word after the other.
BraidStrands -- The strand count $n$ of a braid in $B_n$ .
BraidWrithe -- The exponent sum of a braid word, i.e. the abelianisation $B_n \to \mathbb{Z}$ — and the writhe of the closed diagram.
BurauMatrix -- The reduced Burau matrix of a braid, an $(n-1)\times(n-1)$ matrix over $\mathbb{Z}[t,t^{-1}]$ whose determinant gives the AlexanderPolynomial.
FigureEightKnot -- The figure-eight knot — TwistKnot $(1)$ under its own name.
JonesPolynomial -- The Jones polynomial $V(t)$ of a knot. A knot named $T(p,q)$ takes the closed form; anything else goes through the TEMPERLEY–LIEB algebra rather than a matrix representation, where each crossing becomes its two smoothings and each closed loop is worth $\delta = -A^2 - A^{-2}$ .
KauffmanBracket -- The Kauffman bracket $\langle L\rangle$ of a knot or link, in $A$ — defined for links too, unlike the Jones polynomial, and not itself an invariant until the writhe is corrected for (see BracketInvariant).
KnotCurve -- The sampled points of a knot's embedding in space — only a torus knot has a parameterisation on file, so any other knot declines rather than guessing one from a braid word.
LorenzBraid -- The braid a closed geodesic of the modular flow draws. By Ghys's theorem the modular knots are exactly the periodic orbits of the Lorenz attractor, and those have a purely combinatorial positive braid.
LorenzCurve -- A trajectory of the Lorenz system at its classic parameters, by fourth-order Runge–Kutta, with the initial settling stretch discarded — the attractor whose periodic orbits are the Lorenz knots.
LorenzPermutation -- How the modular flow permutes a closed geodesic's branch-line points, from its $LR$ word.
ParametricCurve -- A curve as three coordinate expressions in $t$ , compiled and sampled over $[0,\text{span}]$ — the general form KnotCurve is a convenience over.
PositivePermutationBraid -- The unique positive braid realising a permutation in which no pair of strands crosses more than once.
PretzelKnot -- The pretzel knot $P(p,q,r)$ , for odd $p,q,r$ — three twisted bands side by side, named by its closed form since no general braid word is known for the family.
SeifertGenus -- The Seifert genus of a knot: the closed form for a named torus, twist or pretzel knot, otherwise Bennequin's $(c-s+1)/2$ on a positive braid's closure.
TorusBraid -- $(\sigma_1\cdots\sigma_{p-1})^q$ in $B_p$ , whose closure is the torus link $T(p,q)$ .
TorusKnot -- The torus knot $T(p,q)$ , winding $p$ times one way and $q$ the other round a torus — a value carrying both a closed form and, when a word for it exists, the TorusBraid it closes from.
TripNumber -- The trip number of a modular geodesic's $LR$ word — the count of $LR$ corners, and the braid index of the Lorenz link it draws.
TwistKnot -- The twist knot with $n$ half-twists past its clasp, for nonzero integer $n$ — $n=1$ is the figure-eight and $n=-1$ is the trefoil.

Calculus

MaxValue -- The global maximum VALUE of f -- Maximize's value alone, without the witnessing point.
Maximize -- The global maximum of f over the real line (or a simple interval constraint), as {value, {x -> point}} -- an exact, provably global answer, not a local one.
MinValue -- The global minimum VALUE of f -- Minimize's value alone, without the witnessing point.
Minimize -- The global minimum of f over the real line (or a simple interval constraint), as {value, {x -> point}} -- an exact, provably global answer, not a local one.
NMaximize -- A numeric global maximum of f over a BOUNDED interval, as {value, {x -> point}} -- dense sampling plus a golden-section refinement, not an exact answer.
NMinimize -- A numeric global minimum of f over a BOUNDED interval, as {value, {x -> point}} -- dense sampling plus a golden-section refinement, not an exact answer.
NSum -- A numeric series -- a finite range summed directly, or a in b = Infinity accelerated to a tight, checked tolerance.

Catalog

Resource -- Looks up a catalog entry by name and, given further arguments, calls its head with them; a name alone resolves to the head itself.

Collections

AbsoluteTiming -- Evaluates an expression and reports the wall-clock time it took.
AbundantNumbers -- The abundant numbers $12, 18, 20, 24, …$ -- integers whose proper divisors sum past them -- as a lazy indexed collection.
Accumulate -- The running sums of a list.
AchillesNumbers -- The Achilles numbers $72, 108, 200, 288, …$ -- powerful but not a perfect power -- as a lazy indexed collection.
AdditiveEnergy -- The number of quadruples $(a_1,a_2,b_1,b_2)\in A\times A\times B\times B$ with $a_1+b_1=a_2+b_2$ .
AdjacencyGraph -- A Graph built from an n x n 0/1 adjacency matrix.
AdjacencyMatrix -- The vertex-by-vertex adjacency matrix of a Graph, indexed by VertexList order.
AdjacentTranspositionInvolutions -- The permutations of $\{1, …, n\}$ that are products of pairwise disjoint adjacent transpositions — every inversion $\pi(i) > \pi(j)$ has $j = i+1$ — as a lazy indexed family.
All -- Whether every element of a collection satisfies a predicate.
AllOnes -- The constant sequence $1, 1, 1, …$ as a lazy indexed collection.
AlternatingPermutations -- Alternating (zigzag) permutations of $\{1, …, n\}$ : $a_1 < a_2 > a_3 < …$ .
AlternatingSignMatrices -- The $n \times n$ alternating sign matrices — entries in $\{-1, 0, 1\}$ , every row and column summing to 1 with alternating nonzero signs — as a lazy indexed family.
Any -- Whether some element of a collection satisfies a predicate.
Append -- The collection with value added as its last element.
AppendTo -- Appends an element to a bound variable's value, and reassigns it in place.
ArithmeticNumbers -- The arithmetic numbers $1, 3, 5, 6, 7, …$ -- integers whose divisors have an integer mean -- as a lazy indexed collection.
Arrangements -- Ordered selections of $k$ distinct elements from $\{1, …, n\}$ — the $k$ -permutations of $n$ .
Array -- f applied over every point of an n-dimensional index range.
Association -- A keyed collection of rules key -> value.
AssociationThread -- An Association pairing keys with values, positionally.
At -- The element at the given position; a negative index counts from the end.
Attributes -- The Wolfram-named properties compute-engine tracks for a head's operator.
AutomorphicNumbers -- The automorphic numbers $1, 5, 6, 25, 76, 376, …$ : $n$ whose square ends in $n$ itself (base 10).
BallotSequences -- Ballot sequences of length $2n$ : 0/1 words where every prefix has at least as many 1s as 0s.
BaxterPermutations -- The permutations of $\{1, …, n\}$ avoiding the vincular patterns $2\text{-}41\text{-}3$ and $3\text{-}14\text{-}2$ — the "41"/"14" descent or ascent must sit at adjacent positions — as a lazy indexed family.
BellNumbers -- The Bell numbers $1, 1, 2, 5, 15, 52, …$ as a lazy indexed collection.
BinaryBracelets -- Binary strings of length $n$ up to rotation AND reflection — the dihedral-group orbits of $\{0,1\}^n$ , a lazy indexed family.
BinaryNecklaces -- The binary words of length $n$ up to rotation, represented by the lexicographically-least word in each orbit.
BinaryPalindromes -- The binary words of length $n$ that read the same reversed.
BinaryStrings -- The $2^n$ binary strings of length $n$ , in standard binary counting order.
BinaryTree -- The singular-inhabitant constructor for a binary tree, nested: a leaf is 0, a node [left, right].
BinaryTreeParentArray -- The singular-inhabitant constructor for a binary tree as its in-order parent array: entry k is the k-th node's parent, 0 at the root.
BinaryWord -- The singular-inhabitant constructor for a word over $\{0, 1\}$ .
BinaryWords -- The binary strings of length $n$ .
BinaryWordsByWeight -- The binary strings of length $n$ with exactly $k$ ones.
BooleanConvert -- expr rewritten in the given boolean normal form — "DNF" (default), "CNF", or "NNF".
BooleanPermutations -- The Boolean permutations of $\{1, …, n\}$ (Tenner) — those avoiding the patterns $321$ and $3412$ — as a lazy indexed family.
BoxedPlanePartitions -- Plane partitions of any size fitting in an $a \times b \times c$ box — at most $a$ rows, each at most $b$ long, entries at most $c$ — as a lazy indexed family.
CarlitzCompositions -- The compositions of $n$ with no two equal adjacent parts.
CarmichaelNumbers -- The Carmichael numbers $561, 1105, 1729, …$ -- composite Fermat pseudoprimes to every coprime base -- as a lazy indexed collection.
Cases -- The elements of a collection matching a pattern.
CatalanNumbers -- The Catalan numbers $1, 1, 2, 5, 14, 42, …$ as a lazy indexed collection.
Catch -- Evaluates an expression, recovering the value of any Throw inside it.
CellularAutomaton -- t+1 fixed-width generations of an elementary 1-D cellular automaton, from init.
CenteredHexagonalNumbers -- The centered hexagonal numbers $1, 7, 19, 37, …$ — $3n^2-3n+1$ — a hexagon of dots grown ring by ring around a center.
CenteredSquareNumbers -- The centered square numbers $1, 5, 13, 25, …$ — $2n^2-2n+1$ — a square of dots grown ring by ring around a center.
CenteredTriangularNumbers -- The centered triangular numbers $1, 4, 10, 19, …$ — $(3n^2-3n+2)/2$ — a triangle of dots grown ring by ring around a center.
CentralDelannoyNumbers -- The central Delannoy numbers $1, 3, 13, 63, 321, …$ as a lazy indexed collection.
CharacterRange -- Every character from c1 to c2, inclusive, in Unicode code-point order.
CirculantGraph -- The circulant graph on n vertices arranged in a circle, joined by one or more fixed offsets.
CircularPrimes -- The circular primes $2, 3, 5, 7, 11, 13, 17, 31, …$ : primes whose every base-10 digit rotation is also prime.
ClearAttributes -- Turns off one of the Wolfram-named properties Attributes can report.
ClosenessCentrality -- How close a vertex of a Graph is, on average, to every other vertex it can reach.
CograssmannianPermutations -- The permutations of $\{1, …, n\}$ with at most one ascent — the value-complement of a GrassmannianPermutations permutation — as a lazy indexed family.
ColoredPermutations -- Permutations of $\{1, \dots, n\}$ with each position colored by one of $k$ colors — the wreath product $\mathbb{Z}_k \wr S_n$ . Element is a $[\text{image}, \text{colors}]$ pair.
Commonest -- Every element tied for the most frequently occurring in the collection.
CompleteGraph -- The complete graph $K_n$ — every pair of the $n$ vertices joined by an edge.
CompleteKaryTree -- The full (perfect) $k$ -ary tree of the given depth — every level completely filled.
ConnectedComponents -- The strongly-connected components of a Graph, respecting edge direction.
ConnectedPermutations -- Indecomposable (connected) permutations of $\{1, …, n\}$ : no proper prefix's values are exactly $\{1, …, j\}$ .
Contains -- Whether a collection has an element structurally equal to v.
Count -- The number of elements equal to value in the collection.
CousinPrimes -- The (lesser) cousin primes $3, 7, 13, 19, 37, …$ : primes $p$ with $p+4$ also prime.
CubeNumbers -- The perfect cubes $1, 8, 27, 64, …$ — $n^3$ .
CycleDecomposition -- The singular-inhabitant constructor for a permutation in cycle notation with its fixed points kept: each cycle from its least point, cycles in order of those points.
CycleGraph -- The cycle graph $C_n$ — $n$ vertices joined in a ring.
DeficientNumbers -- The deficient numbers $1, 2, 3, 4, 5, 7, …$ -- integers whose proper divisors fall short of them -- as a lazy indexed collection.
DelannoyPaths -- Lattice paths from $(0,0)$ to $(n,n)$ using East, North, and Diagonal steps.
DeleteCases -- list with every top-level element matching pattern removed.
DiagonalMatrix -- A square matrix with list down the k-th diagonal (default the main diagonal), zero elsewhere.
DifferenceDelta -- f(n+1) - f(n), simplified -- the forward difference of a sequence.
DirectedEdge -- A directed edge from u to v — the edge head Graph holds in its edge list.
Do -- Evaluates an expression repeatedly, for its side effects, and returns Nothing.
DoubleFactorialNumbers -- The odd double factorials $1, 3, 15, 105, …$ — $(2n-1)!!$ — perfect matchings of $K_{2n}$ .
DyadicCompositions -- The compositions of $n$ into powers-of-two parts.
DyckPath -- The singular-inhabitant constructor for a Dyck path, as its up/down steps.
DyckPaths -- The Dyck paths of semilength $n$ — balanced up/down words — a lazy family of Catalan many.
DyckPathsByHeight -- Dyck paths of semilength $n$ with maximum height exactly $h$ .
Echo -- Prints a value in passing, and returns it unchanged.
EdgeCount -- How many edges a Graph has.
EdgeList -- The edges of a Graph, as declared, in order.
EigenvectorCentrality -- How well-connected each vertex of a Graph is to OTHER well-connected vertices.
EmirpPrimes -- The emirps $13, 17, 31, 37, 71, …$ : primes whose decimal reversal is a DIFFERENT prime ('prime' spelled backwards).
EvenPermutations -- The even permutations of $\{1, …, n\}$ — the alternating group $A_n$ .
EvenSubsets -- The subsets of $\{1, \dots, n\}$ of even size: half of Subsets, $2^{n-1}$ of them for $n \ge 1$ .
EvilNumbers -- The evil numbers $0, 3, 5, 6, 9, …$ : nonnegative integers with an even number of 1-bits.
Extract -- The part of expr at position pos, or a list of parts at several positions.
FactorialNumbers -- The factorials $1, 2, 6, 24, …$ — $n!$ — the number of permutations of $n$ items.
FactorialPower -- The falling factorial x(x-1)…(x-n+1), n factors, for a concrete-number x.
FibStrings -- Binary words of length $n$ with no two consecutive ones.
FibonacciCompositions -- The compositions of $n$ into parts from $\{1, 2\}$ .
FibonacciNumbers -- The Fibonacci numbers $0, 1, 1, 2, 3, 5, …$ as a lazy indexed collection.
FibonacciPrimes -- The Fibonacci primes $2, 3, 5, 13, 89, 233, …$ : Fibonacci numbers that are themselves prime.
FibonacciWords -- The binary strings of length $n$ with no two consecutive 1s: count is the Fibonacci number $F(n+2)$ .
FindShortestPath -- A shortest path from u to v, respecting edge direction (BFS, unweighted).
FinePaths -- Dyck paths of semilength $n$ with no hills — an elementary up/down step touching the ground on both sides.
Finset -- The singular-inhabitant constructor for a subset of $\{1, \dots, n\}$ , as $n$ and its members.
First -- The first element of a collection.
FirstPosition -- The position of the first occurrence of value, searching every level.
FixedPoint -- Iterates f from x until the result no longer changes.
FixedPointList -- Applies a function repeatedly, collecting every value up to the fixed point.
Flatten -- The collection with all levels of nested lists merged into one.
Fold -- Reduce a collection to one value, combining left to right from a seed.
FoldList -- Every intermediate result of folding f over a list.
FreeQ -- Whether pattern matches nowhere in expr — itself, or any subexpression, at any depth.
FromCharacterCode -- The character(s) with the given Unicode code point(s) — the inverse of ToCharacterCode.
FubiniNumbers -- The Fubini numbers (ordered Bell numbers) $1, 1, 3, 13, 75, 541, …$ as a lazy indexed collection.
Gather -- Group identical elements into sublists, in order of first appearance.
GatherBy -- Group elements by the value of a function, in order of first appearance.
GelfandTsetlin -- Gelfand–Tsetlin patterns — triangular interlacing arrays of $n$ rows with entries in $\{0, …, k\}$ — as a lazy indexed family.
GeometricMean -- The n-th root of the product of n values.
GiugaNumbers -- The Giuga numbers $30, 858, 1722, 66198, …$ -- composite $n$ with $p \mid (n/p - 1)$ for every prime $p \mid n$ -- as an indexed collection of unknown count.
GrandDyckPaths -- Free $\pm 1$ -step paths of length $2n$ starting and ending at height $0$ , with no non-negativity constraint.
Graph -- A graph, as an explicit vertex list and edge list (UndirectedEdge/DirectedEdge) — the data every graph query in this library reads.
GraphCenter -- The vertices of a Graph whose eccentricity equals GraphRadius(g).
GraphDiameter -- The greatest eccentricity among all vertices of a Graph.
GraphDistance -- The length of a shortest u to v path, respecting edge direction.
GraphDistanceMatrix -- All-pairs shortest-path distances of a Graph, as a matrix.
GraphPeriphery -- The vertices of a Graph whose eccentricity equals GraphDiameter(g).
GraphRadius -- The smallest eccentricity among all vertices of a Graph.
GrassmannianPermutations -- The permutations of $\{1, …, n\}$ with at most one descent, as a lazy indexed family.
GrayCodeSubsets -- The subsets of $\{1, \dots, n\}$ in binary-reflected Gray-code order: consecutive subsets differ by exactly one element.
GrayCodes -- The binary words of length $n$ in binary-reflected Gray-code order.
GridGraph -- A rectangular (or higher-dimensional) grid graph.
HankelMatrix -- The square matrix, constant along every anti-diagonal, built from c.
HappyNumbers -- The happy numbers $1, 7, 10, 13, 19, …$ : iterating sum-of-squared-digits reaches 1.
HararyGraph -- The minimum-edge k-connected graph on n vertices (Harary, 1962).
HarmonicMean -- The reciprocal of the mean of the reciprocals.
HarshadNumbers -- The Harshad (Niven) numbers $1, 2, 3, …, 10, 12, 18, …$ : integers divisible by their own digit sum.
HeptagonalNumbers -- The heptagonal numbers $1, 7, 18, 34, …$ — $n(5n-3)/2$ — dots in a heptagon.
HexagonalNumbers -- The hexagonal numbers $1, 6, 15, 28, …$ — $H(n) = n(2n-1)$ — dots in a hexagon.
HighlyCompositeNumbers -- The highly composite numbers $1, 2, 4, 6, 12, 24, …$ -- integers with more divisors than any smaller one -- as a lazy indexed collection.
HilbertMatrix -- The (possibly rectangular) Hilbert matrix, entry (i, j) = 1/(i + j - 1), kept exact.
HypercubeGraph -- The $n$ -dimensional hypercube graph — $2^n$ vertices (bit strings), edges between strings differing in one bit.
IdonealNumbers -- Euler's 65 numeri idonei $1, 2, 3, 4, 5, …, 1848$ -- one class per genus of discriminant $-4n$ -- as an indexed collection of unknown count.
IncidenceMatrix -- The vertex-by-edge incidence matrix of a Graph.
IndexOf -- The position of the first occurrence of value in the collection.
IntegerComposition -- The singular-inhabitant constructor for a composition of an integer, as its parts.
IntegerPartitions -- The partitions of $n$ into positive parts, as a lazy indexed family.
Intersection -- The elements common to all the argument collections.
IsAcyclicGraph -- Whether a Graph contains no cycle.
IsArray -- Tests whether t is a List with a uniform (non-ragged) shape at every depth, any rank, optionally with every leaf satisfying test.
IsBipartiteGraph -- Whether a Graph's vertices split into two sets with every edge crossing between them.
IsCompleteGraph -- Whether a Graph is complete — every pair of vertices joined by an edge.
IsConnectedGraph -- Whether a Graph is connected — a single component; strongly connected when it has directed edges.
IsInteger -- Tests whether x is manifestly an integer.
IsIntervalMember -- Tests whether x falls within the interval, honoring Open bounds.
IsIsomorphicGraph -- Whether two Graph values have the same structure, up to relabelling vertices.
IsLoopFreeGraph -- Whether a Graph has no self-loop.
IsMachineNumber -- Whether expr is an ordinary (not extended-precision) inexact number.
IsMatrix -- Tests whether m is a non-empty, non-ragged rank-2 List of Lists, optionally with every entry satisfying test.
IsMersennePrimeExponent -- Tests whether 2^p - 1 is prime, by the Lucas-Lehmer test.
IsNumeric -- Whether expr denotes a definite numeric quantity, without evaluating it.
IsPathGraph -- Whether a Graph is a simple path — a single chain of vertices, each joined once to the next.
IsSimpleGraph -- Whether a Graph has no self-loop and no parallel (repeated) edge.
IsTreeGraph -- Whether a Graph is a tree — connected, with exactly $|V| - 1$ edges.
IsTrue -- Tests whether expr evaluates to the literal True.
IsVector -- Tests whether list is a rank-1 List, optionally with every element satisfying test.
JacobsthalNumbers -- The Jacobsthal numbers $0, 1, 1, 3, 5, 11, …$ as a lazy indexed collection.
Join -- The concatenation of the argument collections, in order.
KAlmostPrimes -- The $k$ -almost primes -- integers with exactly $k$ prime factors, counted with multiplicity ( $\Omega(n)=k$ ) -- as a lazy indexed family, one collection per $k$ .
KBracelets -- Words of length $n$ over a $k$ -letter alphabet up to rotation and reflection — bracelets over $k$ colours, a lazy indexed family.
KCyclePermutations -- Permutations of $\{1, …, n\}$ with exactly $k$ cycles — row $n$ of the unsigned Stirling first-kind triangle.
KDescentPermutations -- Permutations of $\{1, …, n\}$ with exactly $k$ descents — row $n$ of the Eulerian triangle.
KFreeIntegers -- The $k$ -free integers -- naturals with no prime factor raised to the $k$ -th power or higher -- as a lazy indexed family, one collection per $k$ .
KInversionPermutations -- Permutations of $\{1, …, n\}$ with exactly $k$ inversions — row $n$ of the Mahonian triangle.
KLyndonWords -- The Lyndon words of length $n$ over a $k$ -letter alphabet: aperiodic necklaces, strictly less than every nontrivial rotation.
KNecklaces -- The words of length $n$ over a $k$ -letter alphabet up to rotation, represented by the lexicographically-least word in each orbit.
KPermutations -- The $k$ -permutations of $\{1, \dots, n\}$ : ordered selections of $k$ distinct elements, same family as Arrangements.
KaprekarNumbers -- The Kaprekar numbers $1, 9, 45, 55, 99, 297, …$ : $n$ whose square splits into a left and a (nonzero) right part that sum back to $n$ .
KaryTree -- The k-ary tree on n vertices, in breadth-first (heap) layout — sized by vertex count, not depth.
Key -- An accessor tag — At(assoc, Key(k)) looks up the value for key k in an Association.
LargestPartPartitions -- The partitions of $n$ whose largest part is exactly $m$ .
Last -- The last element of a collection.
LatticePaths -- The monotone lattice paths from $(0,0)$ to $(a,b)$ using unit north/east steps, each written as a $0/1$ step sequence ( $1$ = north). Count $\binom{a+b}{a}$ .
LehmerCodes -- The Lehmer code of every permutation of $\{1, …, n\}$ — inversion tables $(a_0, …, a_{n-1})$ with $0 \le a_i \le n-1-i$ .
Length -- The number of elements in the collection.
LetterNumber -- A letter's 1-based position in the English alphabet -- a → 1, …, z → 26.
Level -- Every subexpression of expr at the given level(s) — level 0 is expr itself.
LineGraph -- The graph whose vertices are the edges of g, joined when the original edges share an endpoint.
LinearRecurrence -- The sequence of a linear recurrence with constant coefficients, from its kernel and initial values.
LittleSchroderNumbers -- The little Schröder numbers (super-Catalan numbers) $1, 1, 3, 11, 45, 197, …$ as a lazy indexed collection.
LogicalExpand -- expr rewritten as a disjunction of conjunctions (DNF), with Implies/Equivalent/Xor/Nand/Nor eliminated.
LucasNumbers -- The Lucas numbers $2, 1, 3, 4, 7, 11, …$ as a lazy indexed collection.
LucasStrings -- CIRCULAR binary words of length $n$ with no two consecutive ones, wraparound included.
LuckyNumbers -- The lucky numbers $1, 3, 7, 9, 13, …$ -- survivors of Ulam's positional sieve -- as a lazy indexed collection.
LukasiewiczPaths -- Łukasiewicz paths: length- $(n+1)$ integer words $a_0, …, a_n$ (each $\ge -1$ ) whose prefix sums stay $\ge 0$ and end at $-1$ .
LyndonWords -- The binary Lyndon words of length $n$ : words strictly less than every one of their nontrivial rotations.
MachinePrecision -- The precision of a machine number, $53 \log_{10} 2 \approx 15.95$ decimal digits.
Map -- The collection where each element has been transformed by $f$ ; with several collections, combines them element-wise (like zip), to the length of the shortest.
MapAt -- f applied to the part of expr at position n, leaving the rest of expr unchanged.
MapIndexed -- f applied to each part together with its position path, a list.
MapThread -- f applied across the CORRESPONDING elements of each list, not each list in turn.
MatchQ -- Whether expr matches pattern, at the top level.
Mean -- The arithmetic average of the elements of the collection.
Median -- The middle value of the collection once sorted.
MersennePrimes -- The Mersenne primes $3, 7, 31, 127, 8191, …$ : primes of the form $2^p - 1$ .
Mode -- The most frequently occurring element of the collection.
Module -- Evaluates an expression with local variables, fresh on every call.
MotzkinNumbers -- The Motzkin numbers $1, 1, 2, 4, 9, 21, …$ as a lazy indexed collection.
MotzkinPathsByPeaks -- Motzkin paths of length $n$ with exactly $k$ peaks — an up-step immediately followed by a down-step.
MovingMap -- f applied to each width-(w+1) window of list, sliding by 1 with no padding.
MultiplicativeEnergy -- The number of quadruples $(a_1,a_2,b_1,b_2)\in A\times A\times B\times B$ with $a_1b_1=a_2b_2$ .
NarcissisticNumbers -- The Armstrong (narcissistic) numbers $1, …, 9, 153, 370, …$ : $n$ equal to the sum of its own digits, each raised to the digit count -- the one PROVEN finite family here.
NeighborhoodGraph -- The induced subgraph on the vertices within distance n of v (underlying graph).
Nest -- f applied n times to x.
NestList -- The list of x, f(x), f(f(x)), ... up to n applications of f.
NestWhile -- Applies a function repeatedly while a test on the current value holds.
NestWhileList -- Like NestWhile, but returns every intermediate value along the way.
NonCrossingCycleSupportPermutations -- The permutations of $\{1, …, n\}$ whose cycles, read as a set partition of $\{1, …, n\}$ , form a non-crossing partition — cycles may hold their elements in any cyclic order — as a lazy indexed family.
NonCrossingMatchings -- The perfect matchings of $\{1, …, 2n\}$ with no two crossing chords — the balanced-parenthesis reading of a Dyck path.
NonCrossingPartitions -- The partitions of $\{1, …, n\}$ with no two blocks whose consecutive-element arcs cross.
NonCrossingPermutations -- The noncrossing permutations of $\{1, …, n\}$ (Biane, Kreweras) — those below the long cycle $(1\,2\,…\,n)$ in absolute order — as a lazy indexed family.
NonCrossingTrees -- The spanning trees on $n+1$ circle-labeled points whose edges, drawn as chords, never cross.
NonNestingMatchings -- The perfect matchings of $\{1, …, 2n\}$ with no two nesting chords.
NonNestingPartitions -- The partitions of $\{1, …, n\}$ with no two blocks whose consecutive-element arcs nest.
NoneTrue -- Whether no element of a collection satisfies a predicate.
Normalize -- v divided by its Euclidean norm -- a unit vector in v's direction.
NumberQ -- True only for an explicit numeric literal — a symbolic constant like Pi is NOT NumberQ.
OctagonalNumbers -- The octagonal numbers $1, 8, 21, 40, …$ — $n(3n-2)$ — dots in an octagon.
OddCompositions -- The compositions of $n$ into odd parts.
OddPartitions -- The partitions of $n$ into odd parts.
OddSubsets -- The subsets of $\{1, \dots, n\}$ of odd size: the other half of Subsets, $2^{n-1}$ of them for $n \ge 1$ .
OdiousNumbers -- The odious numbers $1, 2, 4, 7, 8, …$ : positive integers with an odd number of 1-bits.
Ordering -- The permutation of indices that would sort the collection into increasing order.
Outer -- The generalized outer product: f applied to every pair drawn from two lists.
PadLeft -- Pad a list on the left to length n, or truncate it from the left.
PadRight -- Pad a list on the right to length n, or truncate it from the right.
PadovanSequence -- The Padovan sequence $1, 0, 0, 1, 0, 1, 1, 1, 2, …$ as a lazy indexed collection.
PalindromicCompositions -- The compositions of $n$ that read the same forwards and backwards.
PalindromicPrimes -- The palindromic primes $2, 3, 5, 7, 11, 101, …$ : primes that read the same forwards and backwards in base 10.
PartCountBoundedCompositions -- The compositions of $n$ into at most $k$ parts.
PartSizeBoundedCompositions -- The compositions of $n$ into parts of size at most $k$ .
Partition -- The collection split into consecutive, non-overlapping chunks of length n.
PartitionNumbers -- The integer partition counts $p(n) = 1, 1, 2, 3, 5, 7, …$ as a lazy indexed collection.
PartitionsInBox -- The integer partitions fitting in an $a \times b$ box: at most $a$ parts, each at most $b$ . Count $\binom{a+b}{a}$ .
PartitionsIntoKParts -- The integer partitions of $n$ into exactly $k$ positive parts.
PartitionsMaxPart -- The integer partitions of $n$ with every part at most $m$ .
PascalBinomial -- The binomial coefficient, extended to a negative n so Pascal's identity still holds.
PathGraph -- A path graph — vertices joined in a line.
PellNumbers -- The Pell numbers $0, 1, 2, 5, 12, 29, …$ as a lazy indexed collection.
PentagonalNumbers -- The pentagonal numbers $1, 5, 12, 22, …$ — $P(n) = n(3n-1)/2$ — dots in a pentagon.
PentatopeNumbers -- The pentatope numbers $1, 5, 15, 35, …$ — $\binom{n+3}{4}$ — the 4-simplex figurate numbers.
PerfectNumbers -- The perfect numbers $6, 28, 496, 8128, …$ -- integers equal to the sum of their proper divisors -- as a lazy indexed collection.
PerfectPowerNumbers -- The perfect powers $4, 8, 9, 16, 25, …$ -- integers $a^k$ with $a \ge 2, k \ge 2$ -- as a lazy indexed collection.
PermutationsAsCycles -- Every permutation of $\{1, …, n\}$ in cycle notation, fixed points kept, listed by cycle type.
PermutationsAvoiding123 -- Permutations of $\{1, …, n\}$ avoiding the pattern $1-2-3$ .
PermutationsAvoiding132 -- Permutations of $\{1, …, n\}$ avoiding the pattern $1-3-2$ .
PermutationsAvoiding213 -- Permutations of $\{1, …, n\}$ avoiding the pattern $2-1-3$ .
PermutationsAvoiding231 -- Permutations of $\{1, …, n\}$ avoiding the pattern $2-3-1$ .
PermutationsAvoiding312 -- Permutations of $\{1, …, n\}$ avoiding the pattern $3-1-2$ .
PermutationsAvoiding321 -- Permutations of $\{1, …, n\}$ avoiding the pattern $3-2-1$ .
PerniciousNumbers -- The pernicious numbers $3, 5, 6, 7, 9, …$ : positive integers whose binary popcount is itself prime.
PerrinSequence -- The Perrin sequence $3, 0, 2, 3, 2, 5, 5, 7, …$ as a lazy indexed collection.
PetersenGraph -- The Petersen graph — the standard 10-vertex, 15-edge, 3-regular example graph.
PhylogeneticTrees -- The rooted binary trees on $n$ labeled leaves with unlabeled internal nodes, as a lazy indexed family.
Pick -- Elements of list whose corresponding element of sel matches patt (default True).
PlanePartitions -- Plane partitions of $n$ — 2D arrays of positive integers nonincreasing along rows and down columns, summing to $n$ — as a lazy indexed family.
PolygonalNumbers -- The $k$ -gonal figurate numbers $P(k, n) = \big((k-2)n^2 - (k-4)n\big)/2$ , one lazy indexed collection per polygon size $k \ge 3$ .
Position -- Every position value occurs at in the collection.
PowerfulNumbers -- The powerful numbers $1, 4, 8, 9, 16, …$ -- integers where every prime factor appears squared or more -- as a lazy indexed collection.
PowersOfTwo -- The powers of two $1, 2, 4, 8, …$ — subsets of an $n$ -set; row-sums of Pascal's triangle.
PracticalNumbers -- The practical numbers $1, 2, 4, 6, 8, 12, …$ -- integers whose divisors let every smaller integer be written as a distinct-divisor sum -- as a lazy indexed collection.
Precision -- How many significant decimal digits a number carries.
Prepend -- The collection with value added as its first element.
PrimeCompositions -- The compositions of $n$ into prime parts.
PrimeNumbers -- The prime numbers $2, 3, 5, 7, 11, …$ as a lazy indexed collection, unranked by position.
PrimePairs -- The lesser prime $p$ of a pair $(p, p+gap)$ , both prime -- $gap$ selects the family ( $2$ =twin, $4$ =cousin, $6$ =sexy, …), as a lazy indexed family.
PrimePartitions -- The partitions of $n$ into prime parts.
PrimePowerNumbers -- The prime powers $2, 3, 4, 5, 7, 8, 9, 11, …$ : $p^k$ for a prime $p$ and $k \geq 1$ (excluding 1, which is $p^0$ ).
PrimitiveBinaryStrings -- Aperiodic binary strings of length $n$ — words with no proper period — a lazy indexed family (OEIS A027375).
PrimorialNumbers -- The primorials $2, 6, 30, 210, …$ — $p_n\#$ — the product of the first $n$ primes.
Product -- The product of all elements in the collection.
PronicNumbers -- The pronic (oblong) numbers $2, 6, 12, 20, …$ — $n(n+1)$ — twice a triangular number.
ProperCompositions -- The compositions of $n$ into parts $\ge 2$ .
RandomComplex -- A uniformly random complex number in a rectangle of the complex plane.
RandomGraph -- A uniformly-chosen simple undirected Graph with n vertices and exactly m edges.
RandomInteger -- A uniform random integer, or a list (or array) of them -- Random over a Range, by its Wolfram name.
ReIm -- {Re(z), Im(z)} — threads over a list the way Wolfram's Listable ReIm does.
Reap -- Evaluates an expression together with everything Sow recorded during it.
RecurrenceTable -- A table of the values of a sequence defined by a recurrence equation and initial conditions.
Replace -- expr rewritten by lhs -> rhs, but ONLY if the whole expression matches — unlike ReplaceAll, this never looks inside subexpressions.
ReplacePart -- expr with the operand at position i replaced by new — Wolfram 1-based positions, negative counting from the end.
RestrictedGrowthString -- The singular-inhabitant constructor for a restricted growth string: a word w with w₁ = 0 and each letter at most one more than every letter before it.
RestrictedGrowthStrings -- The restricted-growth strings of length $n$ : words $w$ with $w_0 = 0$ and $w_i \le 1 + \max(w_0, …, w_{i-1})$ .
Riffle -- Interleave a separator (or a second list) between the elements of a list.
RiordanPaths -- Motzkin paths of length $n$ with no level step taken at height $0$ .
RootedForests -- The forests of rooted labeled trees on $\{1, \dots, n\}$ , each element a length- $n$ parent array (entry $0$ marks a root). Count $(n+1)^{n-1}$ .
RootedUnlabeledTrees -- The rooted trees on $n$ unlabelled nodes, up to isomorphism, as a lazy indexed family.
RoughNumbers -- The $k$ -rough numbers -- positive integers with no prime factor below $k$ -- as a lazy indexed family, one collection per $k$ (SmoothNumbers's mirror image).
SafePrimes -- The safe primes $5, 7, 11, 23, 47, …$ : primes $q$ with $(q-1)/2$ also prime.
Scan -- The running combination of $f$ over a collection, one result per element.
SchroederNumbers -- The (large) Schröder numbers $1, 2, 6, 22, 90, 394, …$ as a lazy indexed collection.
SeedRandom -- Reseeds the random generator RandomInteger draws from.
SemiperfectNumbers -- The semiperfect numbers $6, 12, 18, 20, …$ -- integers equal to a sum of some subset of their proper divisors -- as a lazy indexed collection.
SemiprimeNumbers -- The semiprimes $4, 6, 9, 10, 14, …$ : products of exactly two primes, with multiplicity ( $\Omega(n) = 2$ ).
SemistandardTableaux -- Semistandard Young tableaux of $n$ cells with entries in $\{1, …, k\}$ , summed over every shape $\lambda \vdash n$ , as a lazy indexed family.
SeparablePermutations -- The permutations of $\{1, …, n\}$ avoiding the patterns $2413$ and $3142$ , as a lazy indexed family.
SetAttributes -- Turns on one of the Wolfram-named properties Attributes can report.
SetComposition -- The singular-inhabitant constructor for a composition of a set, as its ordered blocks.
SetMinus -- The elements of set a that are not also in set b.
SetPartition -- The singular-inhabitant constructor for a partition of a set, as its blocks.
SetPartitions -- The partitions of the set $\{1, …, n\}$ into non-empty blocks, a lazy indexed family.
SexyPrimes -- The (lesser) sexy primes $5, 7, 11, 13, 17, …$ : primes $p$ with $p+6$ also prime.
ShiftedStandardTableaux -- Standard tableaux on shifted diagrams of strict partitions of $n$ — row $i$ starting one column right of row $i-1$ — as a lazy indexed family.
SimplePermutations -- The permutations of $\{1, …, n\}$ with no non-trivial interval — no contiguous run of positions, other than a single position or the whole permutation, whose values form a contiguous range — as a lazy indexed family.
SkewPartitions -- Reduced skew shapes $\lambda/\mu$ with $n$ cells — no empty row or column — as a lazy indexed family.
SkewStandardTableaux -- Standard Young tableaux on reduced skew shapes $\lambda/\mu$ , summed over every shape with $n$ cells, as a lazy indexed family.
SmithNumbers -- The Smith numbers $4, 22, 27, 58, 85, …$ : composite integers whose digit sum equals the digit sum of their prime factors (with multiplicity).
SmoothNumbers -- The $k$ -smooth numbers -- positive integers with every prime factor $\le k$ -- as a lazy indexed family, one collection per $k$ .
SmoothPermutations -- The permutations of $\{1, …, n\}$ avoiding the patterns $3412$ and $4231$ — those whose Schubert variety is smooth — as a lazy indexed family.
SophieGermainPrimes -- The Sophie Germain primes $2, 3, 5, 11, 23, …$ : primes $p$ with $2p+1$ also prime.
Sort -- The elements of the collection in increasing order.
SortBy -- The collection sorted by the value of a function on each element.
Sow -- Sends a value to the nearest enclosing Reap, and returns it.
Span -- A range of positions i;;j;;step used as a part specification.
SparseArray -- A dense array built from position → value rules.
SphenicNumbers -- The sphenic numbers $30, 42, 66, 70, 78, …$ : products of three DISTINCT primes ( $\tau=8$ , $\mu=-1$ ).
Split -- Split a list into runs of identical adjacent elements.
SplitBy -- Split a list into runs on which a function is constant.
SquareFreeNumbers -- The squarefree numbers $1, 2, 3, 5, 6, …$ -- integers divisible by no perfect square $> 1$ -- as a lazy indexed collection.
SquareNumbers -- The perfect squares $1, 4, 9, 16, …$ as a lazy indexed collection, unranked by position.
SquarePartitions -- The partitions of $n$ into perfect-square parts.
SquarePyramidalNumbers -- The square pyramidal numbers $1, 5, 14, 30, …$ — $n(n+1)(2n+1)/6$ — stacked squares, partial sums of the square numbers.
SquarefreeSemiprimes -- The squarefree semiprimes $6, 10, 14, 15, 21, …$ : products of two DISTINCT primes ( $\tau=4$ , $\mu=+1$ ).
StandardTableauPairs -- Pairs $(P, Q)$ of same-shape standard Young tableaux with $n$ cells — the RSK codomain — as a lazy indexed family of $n!$ .
StarGraph -- The star graph on $n$ vertices — one centre joined to $n - 1$ leaves.
StarNumbers -- The star numbers $1, 13, 37, 73, …$ — $6n^2-6n+1$ — centered figurate numbers of a six-pointed star.
SternDiatomicSequence -- Stern's diatomic sequence (the fusc function) $0, 1, 1, 2, 1, 3, 2, 3, …$ as a lazy indexed collection.
StirlingPermutations -- Permutations of the multiset $\{1,1,2,2,…,n,n\}$ where everything between the two copies of $i$ exceeds $i$ , a lazy indexed family of $(2n-1)!!$ .
StringLength -- The number of characters in s.
StringTake -- A substring of s — the first/last n characters, or an explicit 1-based range.
SubexcedantSeq -- The singular-inhabitant constructor for a subexcedant sequence, a word whose entry $i$ (from 1) lies in $0 \le a_i \le i - 1$ .
SubexcedantSeqs -- Subexcedant sequences of length $n$ : words $(a_0, …, a_{n-1})$ with $0 \le a_i \le i$ — another factorial-base encoding of permutations.
Subgraph -- The induced subgraph of g on the given vertices.
Subsets -- The power set of $\{1, …, n\}$ — every subset, as a lazy indexed family of $2^n$ .
SubsetsOfSizeAtMost -- The subsets of $\{1, \dots, n\}$ of size at most $k$ , ordered by size then lexicographically within each size — a size-bounded slice of Subsets.
SubsetsWithoutConsecutive -- The subsets of $\{1, \dots, n\}$ containing no two consecutive integers. Count is the Fibonacci number $F(n+2)$ .
Successor -- The element after $x$ in the collection's own order.
SuperabundantNumbers -- The superabundant numbers $1, 2, 4, 6, 12, 24, …$ -- integers whose abundancy $\sigma(n)/n$ exceeds every smaller one's -- as a lazy indexed collection.
Surd -- The real nth root of a real x, staying real for a negative x when n is odd.
Surjection -- The singular-inhabitant constructor for a surjection from positions onto 1..k, as its word.
Switch -- Evaluates the value paired with the first form the expression matches.
SymmetricGroup -- The $n!$ permutations of $\{1, …, n\}$ as a lazy indexed family, in one-line form.
Table -- An alias for Tabulate (the preferred name) that additionally accepts Mathematica-style iterator specs, e.g. Table(i^2, {i, 1, n}), and the equivalent tuple spelling Table(i^2, (i, 1, n)).
Tabulate -- The list of $f$ applied to each index from 1 to $n$ (or a rectangular grid of them).
TernaryGrayCodes -- Length- $n$ words over $\{0,1,2\}$ in base-3 reflected Gray code order, where consecutive words differ by $\pm1$ in exactly one digit.
TetraCompositions -- The compositions of $n$ into parts from $\{1, 2, 3, 4\}$ .
TetrahedralNumbers -- The tetrahedral numbers $1, 4, 10, 20, …$ — $\binom{n+2}{3}$ — stacked triangles, partial sums of the triangular numbers.
Thread -- f applied elementwise across every list-headed argument, other arguments broadcast.
Through -- A list or sum of functions, called with the same arguments, distributed over head h.
Throw -- Aborts evaluation up to the nearest enclosing Catch, carrying a value.
ThueMorseNumbers -- The Thue–Morse sequence $0, 1, 1, 0, 1, 0, 0, 1, …$ as a lazy indexed collection.
ToCharacterCode -- The Unicode code point of each character of s, as a list of integers.
ToString -- expr printed as a string — of Epsil, this repo's own syntax, not Wolfram InputForm.
Total -- The sum of the elements of a list; a matrix totals column by column.
TriCompositions -- The compositions of $n$ into parts from $\{1, 2, 3\}$ .
TriStrings -- Binary strings of length $n$ with no run of 3 consecutive 1s, a lazy indexed family counted by the tribonacci recurrence.
TriangularCompositions -- The compositions of $n$ into triangular-number parts.
TriangularNumbers -- The triangular numbers $1, 3, 6, 10, …$ — $T(n) = n(n+1)/2$ — dots in a triangle.
TriangularPartitions -- The partitions of $n$ into triangular-number parts.
TribonacciNumbers -- The tribonacci numbers $0, 0, 1, 1, 2, 4, 7, …$ as a lazy indexed collection.
TuranGraph -- The complete r-partite graph on n vertices with parts as equal in size as possible.
TwinPrimes -- The (lesser) twin primes $3, 5, 11, 17, 29, …$ : primes $p$ with $p+2$ also prime.
UndirectedEdge -- An undirected edge between two vertices — the edge head Graph holds in its edge list.
Union -- The union of the argument collections, sorted and de-duplicated; lists give a list, sets a set.
Unique -- The elements of a collection, each kept once, in first-occurrence order.
UnlabeledFreeTrees -- The free (unrooted) trees on $n$ unlabelled nodes, up to isomorphism.
UntouchableNumbers -- The untouchable numbers $2, 5, 52, 88, …$ -- integers that are no number's aliquot sum -- as a lazy indexed collection.
UpTo -- A count specification meaning at most n, taking fewer when fewer are available.
VertexCount -- How many vertices a Graph has.
VertexDegree -- How many edge-ends meet a vertex — direction-blind (a self-loop counts twice).
VertexEccentricity -- The greatest distance from a vertex to any other vertex of a Graph.
VertexInDegree -- The number of edges of a Graph pointing INTO a vertex.
VertexIndex -- The 1-based position of a vertex in a Graph's VertexList.
VertexList -- The vertices of a Graph, in first-appearance order.
VertexOutDegree -- The number of edges of a Graph leaving a vertex.
VexillaryPermutations -- The permutations of $\{1, …, n\}$ avoiding the pattern $2143$ , as a lazy indexed family.
WeakCompositions -- The weak compositions of $n$ into $k$ parts: ordered sequences of $k$ non-negative integers summing to $n$ , unlike IntegerCompositions where every part must be positive.
WeightedAdjacencyMatrix -- The vertex-by-vertex matrix of edge weights of a weighted Graph, 0 where no edge exists.
WeirdNumbers -- The weird numbers $70, 836, 4030, …$ -- abundant but not semiperfect -- as a lazy indexed collection.
WheelGraph -- The wheel graph on n vertices — a hub joined to every vertex of an (n-1)-cycle rim.
While -- Evaluates an expression repeatedly while a test stays true.
With -- Substitutes fixed values into an expression before evaluating it.
Words -- The strings of length $size$ over a $base$ -letter alphabet $\{1, …, base\}$ .
ZigzagCompositions -- The alternating compositions of $n$ : $a_1 < a_2 > a_3 < …$ or $a_1 > a_2 < a_3 > …$ , either starting direction.

Combinatorial Hopf algebras

NSymR -- The ribbon basis of NSym, and the fundamental basis QSymF of QSym. A composition of $n$ is a subset of $\{1,\ldots,n-1\}$ , so the compositions form a Boolean lattice; these bases come from summing over it and inverting.

Combinatorial collections

BinaryTreeParentArrays -- The binary trees with $n$ nodes, each as its in-order parent array.
BinaryTrees -- The binary trees with $n$ internal nodes, as nested lists: a leaf is $0$ , a node $[L, R]$ .
Derangements -- The permutations of $\{1, \dots, n\}$ with no fixed point.
IntegerCompositions -- The compositions of $n$ : ordered sequences of positive parts summing to $n$ .
Involutions -- The self-inverse permutations of $\{1, \dots, n\}$ (only fixed points and 2-cycles).
KSubsets -- The k-element subsets of $\{1, \dots, n\}$ in lexicographic order (the combinations).
Multisets -- The size-k multisets drawn from $\{1, \dots, n\}$ (combinations with repetition).
Tuples -- The k-tuples over $\{1, \dots, n\}$ : all $n^k$ ordered selections with repetition.

Combinatorial maps

AdinBagnoRoichman -- The bijection of Adin, Bagno and Roichman on the 321-avoiding permutation of the path, read back as a path: it keeps the ups before each peak.
AlternatingComplement -- Every second letter swapped for the other, starting with the second.
ArcRepresentation -- Each position linked to the next in its block, or to itself when last.
AssociatedStandardTableau -- The standard tableau whose j-th column holds the j-th smallest elements of the blocks, in increasing order.
Avoiding132Permutation -- The 132-avoiding permutation of Krattenthaler's bijection, built from the path's area sequence read from the end.
Avoiding312Permutation -- The 312-avoiding permutation of Bandlow and Killpatrick: from the identity, each area-sequence entry swaps neighbouring positions that many times.
Avoiding321Permutation -- The 321-avoiding permutation whose RSK pair is the pair of two-row tableaux read off the first and last halves of the path.
BarnabeiCastronuovo -- The Schützenberger involution that reverses the canonical decompositions of the two halves of the path.
BilleyJockuschStanleyPermutation -- The 321-avoiding permutation with entry (ups through a valley) + 1 at the position of its downs through that valley: the Billey-Jockusch-Stanley bijection.
BinarySearchTree -- The tree built by inserting σ(1), σ(2), ... into an empty binary search tree.
BinarySearchTreeParentArray -- The binary search tree of σ as its parent array: entry v is the value v is inserted under, 0 for the root.
BlockConcatenation -- The permutation that lists the blocks in order, each sorted increasingly.
BlockSizeComposition -- The block sizes of a set partition, its blocks taken in order of their least elements.
BlockSizePartition -- The partition of n a set partition's block sizes form, largest first.
BouncePath -- The bounce path of a Dyck path, bouncing between the path and the diagonal from the far corner.
BulgarianSolitaire -- A move of Bulgarian solitaire: the first column of the Ferrers diagram becomes a new row.
CactusEvacuation -- The permutation with the same insertion tableau and the evacuation of the recording tableau.
CallanSwitch -- Callan's involution: switches the first closer with the lesser of the least closer and the second element of the block of 1.
CarPermutation -- Spot j holds the car that parks there, each car taking the first free spot at or after its preference.
Catabolism -- Remove the first row and column-insert its entries back into the rest, largest first.
ClockwiseTriangulationRotation -- Rotate the triangulation of the (n + 2)-gon the path stands for one step clockwise.
CodeComposition -- The composition whose code is the word with a 1 put in front: each 1 opens a part, of 1 plus the zeros after it.
CodeWord -- The word that writes each part i as a one followed by i - 1 zeros.
Complement -- Each entry replaced by n + 1 minus itself.
ComposeApply -- The fold behind Compose — applies the head names in steps to subject right to left, each through its own declared head so every intermediate value stays a properly typed carrier.
CompositionBouncePath -- The Dyck path that bounces off the diagonal once for each part: a run of that many ups, then as many downs.
ConjugacyClassRepresentative -- The canonical permutation with the same cycle type.
ConjugateAfterCycleType -- The conjugate partition of the cycle type.
ConnectivityWord -- Letter i is 1 when the first i entries are all smaller than the rest: the connectivity set as a binary word.
CoriLeBorgne -- The involution that appends a down step to the path, reverses the word, and takes its one rotation that is a Dyck word followed by a down step.
CounterclockwiseTriangulationRotation -- Rotate the triangulation of the (n + 2)-gon the path stands for one step counterclockwise.
CutOutPartition -- The partition cut out of the staircase by the path: part i is the number of downs before the (n + 1 - i)-th up step.
CutWord -- A composition of n as the binary word of length n - 1 marking where it is cut.
CycleAsOneLineNotation -- The cycles, each written from its least point and in order of those points, read as one word.
CyclePartition -- The orbits, as a set partition of the positions.
CycleType -- The multiset of cycle lengths, as a partition.
CyclicShift -- Rotate the word one place to the left.
DeBruijnMorselt -- The right-branch bijection to binary trees followed by the left-branch bijection back: the de Bruijn-Morselt automorphism of the ordered trees.
DecompositionReverse -- Split a path as U A D B at its first return, and swap the pieces, recursively: U B' D A'.
DecreasingRotation -- Every element lowered by one, cyclically: 1 becomes the largest.
DelestViennot -- The Dyck path of the parallelogram polyomino whose columns are as tall as the path's peaks, by the classical bijection.
DeltaMorphism -- The lengths of the maximal runs of equal parts, left to right.
DemazureProductWithInverse -- The Demazure product π⁻¹ ⋆ π in the hyperoctahedral group.
DescentBottomWord -- Letter v is 1 when value v is a descent bottom: some larger value immediately precedes it.
DescentComposition -- The composition of n cut at the descent positions.
DescentSet -- The positions where the word falls.
DescentTopWord -- Letter v - 1 is 1 when value v is a descent top: some smaller value immediately follows it.
DescentWord -- Letter i is 1 when the word falls between positions i and i + 1: the descent set as a binary word.
DistributionMatchHit -- One result of a distribution match: the collection and statistic whose value histogram matches a target histogram, the chain of maps applied first, and its quality — $q_a$ , the overlap of the two histograms (1 exactly when they are identical), and $q_d$ , its discriminating power.
DualMajorIndexToIntertwining -- The inverse of IntertwiningToDualMajorIndex: read the growth word as an intertwining word and rebuild the blocks.
ElizaldeDeutsch -- Reading the steps in the order first, last, second, second to last, and so on, a step of the image is an up step when its matching step has not yet been read.
Evacuation -- Schützenberger's evacuation (involution): remove the entry 1, slide the rest in, and give the cell that empties the largest label, until nothing is left.
Factorization -- A multiplicative factorization as parallel lists — $\prod \text{bases}_i^{\text{exponents}_i}$ — with bases left as number rather than integer, unlike the prime-specific IntegerFactorization.
FindStatHit -- One result of a FindStat-style statistic search: the collection and statistic that reproduce the values, the chain of maps applied first, and its quality — $q_a$ , the fraction of the submitted pairs it explains, and $q_d$ , its discriminating power (distinct values over objects found).
FlagNegativesToNegatives -- The signs rearranged so the negatives of the image count the flag negatives of the original.
FlagZerosToZeros -- The word with as many zeros as this one, prepended with its reverse complement, has occurrences of 10.
Foata -- Foata's fundamental bijection: cycles rotated to their max, ordered by increasing max, parentheses erased.
FoataHan -- Foata and Han's fundamental transformation: each letter in turn is put on the end of the image so far, which is cut into blocks and rotated.
FoataInversionsToMajor -- The inverse of Foata's major-to-inversions bijection: peel off the last letter, rotate the blocks of the rest back.
FoataMajorToInversions -- Foata's bijection: each letter is put on the end of the image so far, which is cut into blocks, and every block is rotated one place right.
Fraction -- A bare numerator/denominator pair over number, kept unreduced — unlike RationalNumber or FractionalNumber, whose components are integer.
FromPermutation -- The increasing binary tree built by minimum-splitting recursion: the position of the smallest value roots the tree, everything before it recurses to the left, everything after it to the right.
GlaisherFranklin -- The Glaisher-Franklin bijection: even part sizes halved become repeated parts, and repeated odd parts merge into their binary multiples.
GrayNext -- The next word in the Gray code: flip the first letter when the ones are even in number, else the letter after the first one.
GrayPrevious -- The previous word in the Gray code: flip the first letter when the ones are odd in number, else the letter after the first one.
IncreasingCyclePermutation -- The permutation whose cycles are the blocks, each cycle increasing: every element goes to the next larger in its block, the largest to the least.
IncreasingRotation -- Every element raised by one, cyclically: the largest becomes 1.
InitialTableau -- The standard tableau of the shape filled row by row with 1, 2, ..., n.
IntertwiningToDualMajorIndex -- The set partition whose growth word is the intertwining word: a bijection taking the intertwining number to the dual major index.
Inverse -- The inverse permutation: position of each value.
InverseAfterComplementAfterReverse -- Reverse, then complement, then invert.
InverseCyclicShift -- Rotate the word one place to the right.
InverseDelestViennot -- The path whose peaks are the polyomino's columns, for the polyomino the classical bijection gives this path.
InverseFoata -- The permutation whose cycles are the stretches of the word between successive left-to-right maxima, each read as a cycle.
InverseFoataHan -- The inverse of Foata and Han's transformation: the last letter is peeled off and the blocks of the rest are rotated back.
InverseGlaisherFranklin -- The inverse of GlaisherFranklin: repeated parts merge into doubled parts, and a leftover part splits into copies of its odd part.
InverseKrewerasComplement -- cπ⁻¹ with the long cycle c = (1 2 ... n), FindStat's order: position i holds π⁻¹(c(i)).
InversePromotion -- Inverse promotion of the Dyck path's two-row standard tableau, read back as a path.
InverseTableauPromotion -- Inverse promotion: replace the entry 1 by n + 1, slide it out to the rim, and subtract one from every entry.
InverseToricPromotion -- The same swaps as ToricPromotion in the opposite order: n and 1, then n - 1 and n, down to 1 and 2.
InverseWachsWhite -- The set partition joined by the arcs of the rook placement that Wachs and White's bijection gamma assigns to the set partition.
InverseWachsWhiteRho -- The set partition joined by the arcs of the rook placement that Wachs and White's bijection rho assigns to the set partition.
InverseYip -- The set partition joined by the arcs of the rook placement that Yip's bijection psi assigns to the set partition.
InverseZetaMap -- The inverse of the zeta map.
KasraouiZeng -- The Kasraoui-Zeng involution: exchanges the number of crossings with the number of nestings.
KnuthClassRepresentative -- The row reading word of σ's RSK insertion tableau — the canonical word of its Knuth (plactic) class.
KnuthKrattenthaler -- The 321-avoiding permutation whose left-to-right maxima are the path's peaks, its RSK pair, and the first row of P with the second row of Q as ups.
KrattenthalerPermutation -- The 321-avoiding permutation whose left-to-right maxima are the path's peaks: Krattenthaler's bijection.
KrewerasComplement -- π⁻¹c with the long cycle c = (1 2 ... n), FindStat's order: position i holds c(π⁻¹(i)).
LabelingPermutation -- The labeling of the parking function's support Dyck path: the positions of the entries, smallest entry first and equal entries left to right.
LalanneKreweras -- The Lalanne-Kreweras involution: labels of the ups and downs that start a double rise or fall, differenced.
LeftBorderSymmetry -- Reverse the order of the right subtrees hanging off the left border, applying the same symmetry inside each.
LeftRotation -- Rotate the root to the left: the right child becomes the root, the root its left child.
LeftToRightMaximaDyckPath -- The Dyck path whose rises climb to each left-to-right maximum and whose falls run to the next one.
LehmerCodeRotation -- The permutation whose Lehmer code is this one's with every entry one larger, cyclically.
LehmerCodeToMajorCode -- The permutation whose major code is this permutation's Lehmer code.
LongCycleConjugation -- Conjugation by the long cycle (1 2 ... n): each cycle's entries shifted up by one, cyclically.
MajorCodeToLehmerCode -- The permutation whose Lehmer code is this permutation's major code.
MinimalDyckPath -- The shortest Dyck path tracing the shape of the Ferrers diagram.
Mirror -- The left-right symmetry of a binary tree: every node's two subtrees swapped.
NegateSigns -- Every sign reversed: multiplication by the longest element −id of the hyperoctahedral group.
NegativeCycleType -- The lengths of the negative cycles, largest first: those of the underlying permutation with an odd number of negative entries.
NegativesToFlagNegatives -- The signs rearranged so the flag negatives of the image count the negatives of the original.
NoncrossingPartition -- Biane's noncrossing set partition: the up steps whose matching downs lie in one run of downs form a block.
NoncrossingPermutation -- The noncrossing permutation Stump's bijection gives the path: the cycles of its touch points, peeled off round by round.
OddPartsWord -- The binary word marking which parts are odd, the parts taken largest first.
ParallelogramPolyominoDyckPath -- The Dyck path of the partition read as a parallelogram polyomino, one column per part.
PathRowmotion -- Rowmotion of the word as a lattice path: the path whose peaks are the original's valleys with an up step appended.
PeakComposition -- The composition cut at the peaks: entries i that are descents after an ascent.
PeakSet -- The interior positions that rise then fall.
PeaksToValleys -- The path with a valley wherever the original has a peak of height at least 2, each valley two lower than its peak.
Peeling -- The path whose area sequence is this one's with its first entry dropped and the rest lowered by two.
PermutahedronVertex -- The vertex (σ(1), …, σ(n)), or its Loday point, is an ordered tuple of coordinates, while finset is an unordered set of positions (the carrier DescentSet and PeakSet use). Reordering the coordinates gives a different vertex but the same finset, so the map would not be injective. A finset-valued PermutahedronVertex is a type mismatch, not a gap to close. Not yet defined.
PositiveCycleType -- The lengths of the positive cycles, largest first: those of the underlying permutation with an even number of negative entries.
PrimeDyckPath -- The path wrapped in one more up step first and down step last: a path that touches the diagonal only at its ends.
Promotion -- Promotion of the Dyck path's two-row standard tableau, read back as a path.
ReadingTableau -- The recording tableau of the row reading word of the shape filled row by row.
ReadingWord -- The rows read left to right, from the bottom row up, as a permutation.
ReflectParallelogramPolyomino -- Reflect the path's parallelogram polyomino across the diagonal, so the first column becomes the first row.
RemoveFirstPart -- The partition without its largest part: its first row removed.
Restricted -- An anonymous restriction of coll to the elements satisfying predicate, delegating to Filter so the base collection stays lazy — the mechanism every named restriction (Derangements, OddPartitions, …) is built on, minus the name.
Reverse -- The word read backwards.
ReverseComplement -- Reverse, then complement.
RightBranchBinaryTree -- An ordered tree as a binary tree with one fewer node: a vertex's right brother becomes its right child, its first child its left child, and the root goes.
RightBranchOrderedTree -- A binary tree as an ordered tree with one more vertex: a right child becomes its parent's right brother, a left child its parent's first child.
RightRotation -- Rotate the root to the right: the left child becomes the root, the root its right child.
Ringel -- The Ringel permutation of the Nakayama algebra whose Kupisch series the path gives, one size up.
RiseComposition -- The lengths of the maximal runs of up steps, left to right.
Rotation -- Attach a leaf to the root, remove the leftmost leaf, and let the vertex it hung from become the root.
RowDescentsToColumnDescents -- A standard tableau of the same shape whose column descents are the original's row descents.
RowSetPartition -- The set partition whose blocks are the rows of the tableau.
Rowmotion -- Rowmotion of the word as an order ideal of a product of two chains: the stretches between valleys reversed, each valley written 10.
Rsk -- The RSK correspondence: the insertion and recording tableaux, as a pair.
RskInsertion -- The insertion tableau of the RSK correspondence.
RskRecording -- The recording tableau of the RSK correspondence, as its rows.
RskShape -- The common shape of the RSK tableaux: the row lengths, as a partition of n.
RunComposition -- The lengths of the maximal runs of equal letters, left to right.
RunSort -- The permutation with its increasing runs sorted lexicographically.
SignWord -- The signs as a binary word: letter i is 1 when entry i is negative.
SimionSchmidt -- The left-to-right minima stay in place and the other positions take the remaining values, largest first: a 123-avoiding permutation.
SortedPartition -- The partition a composition's parts form, sorted largest first.
StandardPermutation -- The permutation ranking the letters of the word, equal letters from left to right.
Standardization -- The permutation of the same relative order, taking 1 < … < n < −n < … < −1.
SupportDyckPath -- The Dyck path whose area sequence is j minus the j-th smallest entry: the parking function's support.
SwapReturnsAndLastDescent -- An involution exchanging the number of returns with the length of the last descent.
TableauDescentWord -- Letter i is 1 when entry i + 1 sits in a lower row than entry i.
TableauPromotion -- Promotion: replace the largest entry by a hole, slide it to the corner at the origin, and add one to every entry.
TableauShape -- The shape of a standard tableau: its row lengths.
TamariBinaryTree -- A path D = L U R D as the binary tree whose left and right subtrees are those of L and R: the bijection compatible with the Tamari order.
TamariDyckPath -- The Dyck path T(L) U T(R) D of a tree with subtrees L and R: the bijection compatible with the Tamari order.
ToLehmerCode -- Entry i counts the later entries smaller than p(i).
ToPermutation -- The permutation an RSK pair comes from: inverse RSK.
ToricPromotion -- Toric promotion of the path graph: swap the values 1 and 2, then 2 and 3, and so on up to n and 1, each swap skipped when the two values sit next to each other.
TouchComposition -- The semilengths of the primitive pieces between a path's returns to the diagonal.
TransposeComposition -- The conjugate composition: its ribbon shape transposed, the complement of the reverse.
TransposePartition -- The conjugate partition: column i of the Ferrers diagram becomes row i.
TransposeTableau -- The conjugate tableau: rows and columns exchanged.
Twist -- The first letter swapped for the other.
TwoRowTableau -- The standard tableau of shape (n, n) with the positions of the up steps in its first row and of the down steps in its second.
UnderlyingPermutation -- The permutation left when the signs are forgotten.
UnderlyingSetPartition -- The set partition with the same blocks, the order of the blocks forgotten.
ValleyComposition -- The composition cut at the valleys: entries i that are ascents after a descent.
ValleysToPeaks -- Every valley 01 swapped for a peak 10, and a trailing 0 made a 1.
WachsWhite -- The set partition whose arcs, read as a rook placement, Wachs and White's bijection gamma carries to a set partition.
WachsWhiteRho -- The set partition whose arcs, read as a rook placement, Wachs and White's bijection rho carries to a set partition.
WeakExceedancePartition -- The coarsest set partition containing every arc from i to π(i) with i ≤ π(i).
Yip -- The set partition whose arcs, read as a rook placement, Yip's bijection psi carries to a set partition.
ZerosToFlagZeros -- The word whose reverse complement, prepended, has as many occurrences of 10 as this has zeros.
ZetaMap -- The zeta map: the area sequence read backwards, level by level from the top, as ups and downs.

Combinatorics 1 ≝

BellNumber -- The nth Bell number, counting the ways to partition a set of n elements.
Binomial -- The binomial coefficient $\binom{n}{k}$ , the number of k-element subsets of an n-element set.
CatalanNumber -- The nth Catalan number $C_n$ , counting balanced bracket sequences, binary trees, polygon triangulations, and many other structures of size n.
DeleteFromTableau -- Reverse row insertion at the end of row $r$ of a Young tableau, the step inverse RSK repeats.
DiscreteRatio -- The ratio $f(n+1)/f(n)$ between consecutive terms of a sequence, simplified.
Eulerian -- The Eulerian number $A(n, k)$ , the number of permutations of $\{1, …, n\}$ with exactly $k$ descents (equally, $k$ ascents).
ExponentialGeneratingFunction -- The exponential generating function $\sum_n \mathrm{expr}(n)\,x^n/n!$ of a sequence, in closed form, when one exists.
Factorial -- The product of the positive integers up to n, written n!.
Factorial2 -- The double factorial $n!!$ , the product of every second integer from n down to 1 or 2.
FindSequenceFunction -- A closed form $a(n)$ for a sequence given as a list $a(0), a(1), \dots$ , when one can be found.
GeneratingFunction -- The ordinary generating function $\sum_n \mathrm{expr}(n)\,x^n$ of a sequence, in closed form, when one exists.
MahonianNumber -- The number of permutations of $\{1, …, n\}$ with exactly $k$ inversions — entry $k$ of row $n$ of the Mahonian triangle.
Multinomial -- The multinomial coefficient $\frac{(k_1 + k_2 + \cdots)!}{k_1!\,k_2!\cdots}$ generalises Binomial to more than two parts.
Pochhammer -- The rising factorial (a)_n = a(a+1)(a+2)…(a+n−1).
RisingFactorial≝ -- The rising factorial $a(a+1)\cdots(a+n-1)$ : Fungrim's name for Pochhammer, which it delegates to, extended to a complex non-integer order. Provided by @enumeratio/analytic.
Stirling -- The Stirling number of the second kind $S(n, k)$ : the number of ways to partition an n-set into k non-empty blocks
StirlingS1 -- The signed Stirling number of the first kind, relating falling factorials to powers of n.
Subfactorial -- The number of derangements of n objects: permutations that leave no element in its original position.

Composition statistics

AbsoluteVariation -- The sum of |c(i+1) - c(i)|.
AscentVariation -- The sum of c(i+1) - c(i) over the positions where it is positive.
AsymmetryDegree -- Pairs of mirror positions holding different parts.
BargraphHalfPerimeter -- Up steps plus horizontal steps of the bargraph.
BargraphPeakCount -- Peaks of the bargraph with bars of these heights: a rise, level steps, then a fall.
BargraphUpStepCount -- Up steps of the bargraph.
BargraphValleyHeightSum -- The sum of the heights of the valleys of the bargraph.
CompositionMajorIndex -- The sum of the descents of the composition: its partial sums, the total left out.
CompositionNakayamaGlobalDimension -- The global dimension of the linear Nakayama algebra whose Kupisch series is c_1 + 1, ..., 2, c_2 + 1, ..., 2, ..., 1.
ConstantRunCount -- Maximal runs of equal parts.
DualImmaculateMonomialColumnSum -- The sum of the entries in the column of the composition in the matrix from the dual immaculate to the monomial quasisymmetric functions.
DyckPathsAboveBouncePathCount -- Dyck paths of semilength n that stay weakly above the bounce path of the composition: their area sequences are at least 0 .. c_1 - 1, 0 .. c_2 - 1, ....
FirstPart -- The first part.
InverseDescentParkingFunctionCount -- Parking functions of size n whose diagonal reading word has the inverse descent composition c.
LastPart -- The last part.
LongestConstantRun -- The length of the longest run of equal parts.
LongestStaircase -- The greatest l with parts at positions i_1 < ... < i_l such that part i_m is at least m.
LongestWeaklyIncreasingSubsequence -- The length of a longest weakly increasing subsequence of the parts.
PartsEqualToReversedPart -- Positions i where the part equals the part at the mirror position.
RibbonCornerCount -- Corners of the ribbon: its inner and outer corners together.
SeaweedIndex -- Twice the cycles plus the paths, less one, of the meander of the composition over the single block: the index of the maximal parabolic seaweed algebra.
ShortestConstantRun -- The length of the shortest run of equal parts.
SmallestMultiplicity -- The smallest number of parts of one size, among the sizes that occur.
SmallestPart -- The smallest part.
StandardCompositionTableauCount -- Standard composition tableaux of the shape: fillings of 1 .. n with rows decreasing, the first column increasing, and the triple rule.
StandardImmaculateTableauCount -- Standard immaculate tableaux of the shape: fillings of 1 .. n increasing along the rows and down the first column.
StandardRibbonTableauCount -- Standard fillings of the ribbon whose rows have these lengths: the permutations of 1 .. n with these partial sums as their descents.
StrongRecordCount -- Parts greater than every part before: the first part and each new maximum.
StrongRecordPositionSum -- The sum of the positions of the strong records.
Variation -- The signed sum of c(i+1) - c(i): the last part less the first.
WeakDescentCount -- Positions i with c(i) >= c(i+1).
WeakMajorIndex -- The sum of the positions i with c(i) >= c(i+1).
WeakRecordCount -- Parts at least as great as every part before.
WeakRecordPositionSum -- The sum of the positions of the weak records.

Compute engine

About -- What the system knows about a name, as a dictionary.
Apply -- Apply a function to a list of arguments.
D -- The symbolic partial derivative of an expression with respect to one or more variables.
Derivative -- The derivative operator: the function that is the $n$ th derivative of $f$ .
Erfi -- The imaginary error function $\operatorname{erfi}(z) = -i\operatorname{erf}(iz)$ .
FromContinuedFraction -- The value of a continued fraction given its list of terms $[a_0, a_1, \ldots]$ .
Imaginary -- Imaginary part of a complex number.
Integrate -- The symbolic integral of an expression, with optional bounds.
Limit -- The limit of $f$ as its argument approaches $point$ , optionally from one side.
MatrixPower -- Compute-engine's own MatrixPower, widened to accept a ModularMatrix or a word in place of an ordinary matrix.
NIntegrate -- The numeric integral of a function between two bounds.
Permutations -- Return all permutations of length k (default full length) of a collection.
Primes -- The set of prime numbers, a constant compute-engine declares; it answers membership, not position.
Random -- A random element of a domain: uniform over a finite collection or an interval, or drawn from a distribution; with a count or shape, that many draws.
Re -- The real part of a complex number. An alias for Real, the preferred name.
Real -- Real part of a complex number.
Simplify -- A simpler equivalent of an expression, optionally under assumptions such as $x > 0$ .
Solve -- The list of solutions of an equation, or a system of equations, for its unknown(s).
TrigToExp -- Trigonometric and hyperbolic functions rewritten exactly in terms of the complex exponential.

Controlling evaluation

MemoryConstrained -- Evaluates expr under a memory cap of bytes — real only in the isolated evaluator.
TimeConstrained -- Evaluates expr, but aborts after t seconds and returns failexpr.
VerificationTest -- Evaluates input and compares it with expected, returning a TestResultObject.

Diagram algebras

BrauerAlgebra -- $B_n(\delta)$ : the diagrams that are perfect matchings — every point paired with exactly one other. Dimension $(2n-1)!!$ .
Diagram -- A diagram: a set partition of $2n$ points, a top row $1…n$ and a bottom row $1'…n'$ , written as blocks of signed labels. The basis element of every algebra below.
DiagramCoarsenings -- Every partition coarser than a diagram's — $\mathrm{Bell}(k)$ of them, for a diagram with $k$ blocks.
InDiagramBasis -- Rewrite a partition-algebra element in the diagram basis $d_\lambda$ , by Möbius inversion over the partition lattice when it arrived in the orbit basis.
InOrbitBasis -- Rewrite a partition-algebra element in the orbit basis $x_\lambda$ , which asks for exactly those points to be connected and nothing else.
MotzkinAlgebra -- $M_n(\delta)$ : the planar diagrams whose blocks have size at most two — so points may also be left unpaired. Dimension $M(2n)$ , the Motzkin numbers.
OrbitDiagram -- The orbit basis of the partition algebra. A diagram $d_\lambda$ asks for points to be connected; an orbit element $x_\lambda$ asks for them to be connected AND NOTHING ELSE — so the two bases differ by Möbius inversion over the partition lattice.
PartitionAlgebra -- $P_n(\delta)$ : every set partition of $2n$ points. The largest of the family — all the others are subalgebras cut out by admitting fewer diagrams. Dimension $B(2n)$ , the Bell numbers.
TemperleyLiebAlgebra -- $TL_n(\delta)$ : the PLANAR perfect matchings — the diagrams you can draw in a rectangle without crossings. Dimension $C(n)$ , the Catalan numbers.

Dyck path and binary word statistics

SymmetryDegree -- The positions, counting inward from the ends, at which a Dyck path or a binary word agrees with its mirror image.

Dyck path statistics

AdjointInvariantDescents -- The descents of the invariant in a tensor power of the adjoint representation of GL2 that the path indexes, read over its step pairs.
Area -- The whole squares between the path and the diagonal below it: the sum of the area sequence.
AreaAboveBouncePath -- The boxes between the path and its bounce path: the area less the area of the bounce path.
BicoloredMotzkinHeight -- The height of the bicolored Motzkin path made by reading the steps in pairs: two up steps rise, two down steps fall.
BicoloredMotzkinLevelSteps -- The level steps of that bicolored Motzkin path: pairs of steps with one up and one down.
Bounce -- The bounce statistic: the sum of the bounce points of the path that bounces from its end back toward the origin.
BounceCount -- The number of bounce points of the bounce path, not counting the two ends.
BounceDeficit -- The binomial(n, 2) less the area and the bounce.
BoxWeight -- The product, over the up steps, of one more than the number of down steps before it.
BoxesBelowAtLeastTwoPeaks -- The boxes between the path and the diagonal that lie right of and below at least two peaks.
BoxesBelowOnePeak -- The boxes between the path and the diagonal that lie right of and below exactly one peak.
CenteredMultitunnels -- Factorizations of the path into three parts, the middle one a Dyck path and the outer two of equal length.
CenteredTunnels -- Tunnels (an up step with its matching down step) centered on the middle of the path: positions summing to the length plus one.
Coarea -- The complement of the area within the staircase: binomial(n, 2) less the area.
DiagonalStandardMonomialDegree -- The degree of the standard monomial of the path relative to the diagonal boundary.
Dinv -- The dinv statistic, read from the area sequence.
DinvDeficit -- The binomial(n, 2) less the area and the dinv: the boxes of the partition traced out by the path where leg minus arm is not 0 or 1.
DoubleFallPositionSum -- The sum of the positions where a down step is followed by a down step.
DoubleRisePositionSum -- The sum of the positions where an up step is followed by an up step.
DoubleRises -- Occurrences of two consecutive up steps.
DoubleStepVisiblePoints -- Points below the path, at the parity of the path's height there, from which the north-east diagonal meets a double fall and the north-west diagonal meets a double rise.
DownStepHeightProduct -- The product, over the down steps, of the height each starts from.
DownStepMajorIndex -- The major index counting positions in down steps only: over each valley, the number of down steps up to and including its down step; summed.
DualRegularModuleFirstExtensionDimension -- The dimension of the first extensions of the direct sum of the indecomposable injective modules by the regular module.
FirstComponentSemilength -- The position of the first return to height 0, counted in up steps: half the steps of the first primitive component.
FirstDownStepPosition -- The position of the first down step.
FirstMaximalPeakPosition -- The position of the up step into the first peak of maximal height.
FirstSimpleOfMaximalProjectiveDimension -- The first vertex, counting from 0, whose simple module has projective dimension equal to the global dimension.
FirstSimpleProjectiveDimension -- The projective dimension of the first simple module of the path's linear Nakayama algebra.
GlobalMaxima -- The number of points of the path at the greatest height it reaches.
HalfEvenSimpleProjectiveDimensionSum -- Half the sum of the projective dimensions of the simple modules that have even projective dimension.
HalfMacMahonEqualIndex -- Half the sum of the positions of double rises and double falls.
Hills -- Peaks at height 1 — an up step from the axis immediately followed by a down step.
HillsOfSizeThree -- Occurrences of three up steps then three down steps, starting on the axis.
HillsOfSizeTwo -- Occurrences of two up steps then two down steps, starting on the axis.
InitialRise -- The length of the opening run of up steps.
InjectiveProjectiveDimensionSum -- The sum of the projective dimensions of the indecomposable injective modules of the path's linear Nakayama algebra.
InjectivesOfCodominantDimensionAtLeastTwo -- The indecomposable injective modules of codominant dimension at least two.
InjectivesOfCodominantDimensionEqualToGlobalDimension -- The indecomposable injective modules whose codominant dimension is the global dimension.
InjectivesOfEvenProjectiveDimension -- The indecomposable injective modules of even projective dimension.
InjectivesOfMaximalProjectiveDimension -- The indecomposable injective modules whose projective dimension is the global dimension.
InjectivesOfNonMaximalSocleProjectiveDimensionMinusProjectiveInjectives -- The indecomposable injective modules whose socle has projective dimension less than the global dimension, less the indecomposable projective-injective modules.
InjectivesOfNonVanishingSecondExtensionWithRegularModule -- The indecomposable injective modules with a nonzero second extension by the regular module.
InjectivesOfProjectiveDimensionAtMostOne -- The indecomposable injective modules of projective dimension at most one.
InjectivesOfProjectiveDimensionAtMostTwo -- The indecomposable injective modules of projective dimension at most two.
InjectivesOfProjectiveDimensionOne -- The indecomposable injective modules of projective dimension one.
InjectivesOfProjectiveDimensionOneBelowMaximal -- The indecomposable injective modules whose projective dimension is one less than the global dimension.
InjectivesOfProjectiveDimensionTwo -- The indecomposable injective modules of projective dimension two.
InteractionDifference -- The pyramid weight less the upper interactions: lower interactions less upper interactions.
InteriorReturns -- Returns to height 0 strictly before the end.
LargestAreaSequenceEntry -- The largest entry of the area sequence: the greatest area to the right of an up step.
LastDoubleRisePosition -- The position of the last double rise; 0 if there is none.
LastSimpleOfMaximalProjectiveDimension -- The last vertex, counting from 0, whose simple module has projective dimension equal to the global dimension.
LastUpStepPosition -- The position of the last up step.
LeftTunnels -- Tunnels (an up step with its matching down step) left of the middle: positions summing to less than the length plus one.
LogarithmicHeight -- The floor of the binary logarithm of the height plus one.
LongTunnels -- The longest sequences of consecutive tunnels: the semilength less the valleys.
LongestAscent -- The longest run of consecutive up steps.
LongestDescent -- The longest run of consecutive down steps.
LongestZigzag -- The length of the longest stretch of steps that alternate between up and down.
MacMahonEqualIndex -- The sum of the positions of double rises and double falls: MacMahon's equal index of the step word.
MaximalFirstExtensionWithRegularModule -- The greatest dimension of the first extensions of an indecomposable module by the regular module.
MaximalModuleProjectiveAndInjectiveDimensionSum -- The greatest sum of the projective and the injective dimension of an indecomposable module.
MaximalSecondExtensionOfSimplesWithRegularModule -- The greatest dimension of the second extensions of a simple module by the regular module.
MaximalSimpleDominantDimension -- The greatest dominant dimension of a simple module.
MaximalSimpleProjectiveAndInjectiveDimensionSum -- The greatest sum of the projective and the injective dimension of a simple module.
MiddleNonRunHeight -- The height of the middle turning point: the middle one of the peaks and valleys, left to right.
MinimalPeakHeight -- The height of the lowest peak.
MinimalRise -- The length of the shortest rise.
MinimalValleyHeight -- The lowest height of a valley, or the semilength if the path has none.
ModulesOfMaximalProjectiveAndInjectiveDimension -- The indecomposable modules whose projective and injective dimensions both equal the global dimension.
ModulesOfMaximalProjectiveDimension -- The indecomposable modules whose projective dimension is the global dimension.
ModulesOfProjectiveAndInjectiveDimensionAtLeastTwo -- The indecomposable modules whose projective and injective dimensions are both at least two.
ModulesOfProjectiveAndInjectiveDimensionAtMostOne -- The indecomposable modules whose projective and injective dimensions are both at most one.
ModulesOfProjectiveDimensionAtMostOne -- The indecomposable modules of projective dimension at most one.
ModulesOfProjectiveDimensionTwo -- The indecomposable modules of projective dimension two.
ModulesOfProjectiveOrInjectiveDimensionAtMostOne -- The indecomposable modules whose projective dimension or injective dimension is at most one.
ModulesOfVanishingFirstExtensionWithRegularModule -- The indecomposable modules with no first extension by the regular module.
NakayamaDominantDimension -- The dominant dimension of the path's linear Nakayama algebra.
NakayamaGlobalDimension -- The global dimension of the linear Nakayama algebra of the path: the largest projective dimension of a simple module.
NakayamaGlobalMinusDominantDimension -- The global dimension less the dominant dimension of the path's linear Nakayama algebra.
NestedCenteredTunnelPairs -- Pairs of centered tunnels, one strictly inside the other.
NestedLeftTunnelPairs -- Pairs of left tunnels, one strictly inside the other.
NonFinalConstantRunsOfLengthAtLeastTwo -- Maximal runs of equal steps, at least two long, that are followed by a step of the other kind: the occurrences of two up steps then a down step, or two down steps then an up step.
NonProjectiveInjectiveCodominantDimensionSum -- The sum of the codominant dimensions of the non-projective indecomposable injective modules.
NonProjectiveInjectiveProjectiveDimensionSpread -- The difference between the greatest and the least projective dimension of a non-projective indecomposable injective module.
NonProjectiveInjectiveToNonInjectiveProjectiveHomDimension -- The dimension of the homomorphisms from the direct sum of the non-projective indecomposable injective modules to the direct sum of the non-injective indecomposable projective modules.
NoncrossingMergeableSingletonPairs -- Pairs of singleton blocks of the path's noncrossing partition that can be merged into one block, leaving the partition noncrossing.
NorthStepOrderingCount -- The total orderings of the up steps such that the up steps after the k-th down step are never among the first k in the order.
OccurrencesOfDDU -- Occurrences of two down steps followed by an up step.
OccurrencesOfUDU -- Occurrences of an up, a down and an up step.
OccurrencesOfUUU -- Occurrences of three consecutive up steps.
ParallelogramPolyominoArea -- The cells of the parallelogram polyomino the path corresponds to under the Delest-Viennot bijection.
ParallelogramPolyominoInnerCorners -- The inner corners of the parallelogram polyomino: the turns from east to north on its two bounding paths, not counting the corners every polyomino has.
ParallelogramPolyominoMaximalColumnHeight -- The height of the tallest column of the parallelogram polyomino.
ParallelogramPolyominoMinimalColumnHeight -- The height of the shortest column of the parallelogram polyomino.
ParallelogramPolyominoNormalizedArea -- The area of the parallelogram polyomino less the semilength: the cells beyond the smallest polyomino.
ParallelogramPolyominoUpperInnerCorners -- The inner corners of the polyomino's upper path: its turns from east to north, not counting the first.
ParkingFunctionCount -- The parking functions supported by the path: semilength factorial over the product of the factorials of the rise lengths.
PeakHeightExcess -- The sum of the heights of the peaks, less the number of peaks.
PeakHeightProduct -- The product of the heights of the peaks.
PeakHeightSquareSum -- The sum of the squares of the heights of the peaks.
PeakHeightSum -- The sum of the heights of the peaks.
PeakLeftToRightMaxima -- Peaks higher than every peak to their left.
PeakRectangleAreaSum -- The sum, over consecutive peaks, of the area of the rectangle they span with the valley between them.
PeaksMinusNakayamaDominantDimension -- The peaks of the path less the dominant dimension of its linear Nakayama algebra.
PeaksMinusNakayamaGlobalDimension -- The peaks of the path less the global dimension of its linear Nakayama algebra.
ProjectiveMinusCodominantDimensionSum -- The sum, over the non-projective indecomposable injective modules, of their projective dimension less their codominant dimension.
ProjectivesOfMaximalInjectiveDimension -- The indecomposable projective modules whose injective dimension is the global dimension.
ProperlyAboveDyckPathCount -- The Dyck paths of the same semilength weakly above this path, itself excluded.
ProperlyBelowDyckPathCount -- The Dyck paths of the same semilength weakly below this path, itself excluded.
PyramidWeight -- Over each peak, the shorter of the up run before it and the down run after it, summed: the sizes of the maximal pyramids.
RadicalEndomorphismDimension -- The dimension of the endomorphisms of the Jacobson radical of the path's linear Nakayama algebra.
RadicalSelfExtensionDimension -- The dimension of the first extensions of the Jacobson radical by itself.
RadicalToSquaredRadicalExtensionDimension -- The dimension of the first extensions of the Jacobson radical by its square.
RegularModuloSocleToRadicalExtensionDimension -- The dimension of the first extensions of the regular module modulo its socle by the Jacobson radical.
Returns -- Points where the path comes back to height 0.
RisesOfLengthAtLeastThree -- Rises (maximal runs of up steps) of length at least 3.
RisesOfLengthAtLeastTwo -- Rises (maximal runs of up steps) of length at least 2.
RisesOfLengthOne -- Rises (maximal runs of up steps) of length 1.
RisesOfLengthOneAtOddHeight -- Rises of length 1 that start at an odd height.
RisesOfLengthThree -- Rises (maximal runs of up steps) of length 3.
RisesOfLengthTwo -- Rises (maximal runs of up steps) of length 2.
RunsOfLengthOne -- Maximal runs of equal steps that are a single step.
SecondToLastInjectiveProjectiveDimension -- The projective dimension of the indecomposable injective module whose socle is the second-to-last simple module.
SetValuedTableauComajorIndex -- The comajor index of the set-valued two-row tableau the path corresponds to: the sum, over its natural descents k, of the semilength plus one less k.
SetValuedTableauNaturalDescents -- The natural descents of the set-valued two-row tableau the path corresponds to, read off the step pairs by FindStat's rules.
SimpleProjectiveDimensionProduct -- The product of the nonzero projective dimensions of the simple modules of the path's linear Nakayama algebra.
SimpleProjectiveDimensionSum -- The sum of the projective dimensions of the simple modules of the path's linear Nakayama algebra.
SimpleSummandsOfSecondRadicalLayer -- The simple summands of J^2/J^3 for the Jacobson radical J of the path's linear Nakayama algebra, one for each indecomposable projective of dimension at least three.
SimplesOfDominantDimensionAtLeastThree -- The simple modules whose dominant dimension is at least three.
SimplesOfDominantDimensionEqualToGlobalDimension -- The simple modules whose dominant dimension is the global dimension.
SimplesOfEqualProjectiveAndInjectiveDimension -- The simple modules whose projective dimension equals their injective dimension.
SimplesOfEvenProjectiveDimension -- The simple modules of even projective dimension.
SimplesOfMaximalEvenProjectiveDimension -- The simple modules whose projective dimension is the greatest even one that occurs: the global dimension if it is even, else one less.
SimplesOfMaximalProjectiveDimension -- The simple modules whose projective dimension is the global dimension.
SimplesOfNonMaximalProjectiveDimension -- The simple modules whose projective dimension is less than the global dimension.
SimplesOfProjectiveDimensionAtMostFour -- The simple modules of projective dimension at most four.
SimplesOfProjectiveDimensionAtMostThree -- The simple modules of projective dimension at most three.
SimplesOfProjectiveDimensionAtMostTwo -- The simple modules of projective dimension at most two.
SimplesOfProjectiveDimensionFour -- The simple modules of projective dimension four.
SimplesOfProjectiveDimensionThree -- The simple modules of projective dimension three.
SimplesOfProjectiveDimensionTwo -- The simple modules of projective dimension two.
SimplesOfSmallInjectiveDimensionOrPositiveDominantDimension -- The simple modules whose injective dimension is at most one or whose dominant dimension is at least one.
SimplesOfTwoDimensionalSecondExtensionWithRegularModule -- The simple modules whose second extensions by the regular module have dimension two.
SimplesOfVanishingSecondExtensionWithRegularModule -- The simple modules with no second extension by the regular module.
SkewHookPositionSum -- The sum of the skew hook positions: after two up steps and a down step, the down step's rank among the down steps; after two down steps and an up step, the rank of the second down step.
SocleSimplesOfVanishingFirstExtensionWithRegularModule -- The simple modules in the socle of the regular module that have no first extension by the regular module.
SocleToRadicalExtensionDimension -- The dimension of the first extensions of the socle of the regular module by the Jacobson radical.
ThreeCatalanPathCount -- The paths from the origin to (n, n, n) with x >= y >= z whose first two coordinates trace this Dyck path and whose last two coordinates trace it as well.
ThreeDimensionalModulesOfProjectiveDimensionOne -- The indecomposable modules of dimension three and projective dimension one.
TouchPointCount -- Points where the path touches the axis — the returns.
TwoDimensionalModulesOfProjectiveDimensionOne -- The indecomposable modules of dimension two and projective dimension one.
UnitIntervalGraphTwinVertices -- The twin vertices of the path's unit interval graph: the indices where the next up step and the next down step each follow directly.
UpStepMajorIndex -- The major index counting positions in up steps only: over each valley, the number of up steps up to its down step; summed.
UpStepsAfterLastDoubleRise -- Up steps after the first step of the last double rise; all of them if there is none.
UpStepsAtEvenPositions -- Up steps at an even position, counting from 1.
UpStepsAtOddPositions -- Up steps at an odd position, counting from 1.
UpperInteractions -- Over each valley, the shorter of the down run before it and the up run after it, summed: the factors of k down steps then k up steps.
ValleyPositionSum -- The sum of the positions of the valleys: a down step followed by an up step.
ValleyTunnelSemilengthSum -- The sum of the semilengths of the tunnels that end at a valley.
ValleysAboveAxis -- Valleys not on the axis: a down step followed by an up step, at height above 0.
WeakPeakLeftToRightMaxima -- Peaks at least as high as every peak to their left.
WeaklyAboveDyckPathCount -- The Dyck paths of the same semilength weakly above this path, itself included.
WeaklyAboveLatticePathCount -- The up-and-down paths of the same length, ending anywhere, that stay weakly above this path.
WeaklyBelowDyckPathCount -- The Dyck paths of the same semilength weakly below this path, itself included.

Elementary functions 1 primitive

Arccos -- Arccosine, the inverse of Cos restricted to $[0, \pi]$ .
Arccot -- Arccotangent, the inverse of Cot.
Arccsc -- Arccosecant, the inverse of Csc.
Arcoth -- Inverse hyperbolic cotangent, the inverse of Coth.
Arcsch -- Inverse hyperbolic cosecant, the inverse of Csch.
Arcsec -- Arcsecant, the inverse of Sec.
Arcsin -- Arcsine, the inverse of Sin restricted to $[-\pi/2, \pi/2]$ .
Arctan -- Arctangent, the inverse of Tan restricted to $(-\pi/2, \pi/2)$ .
Arsech -- Inverse hyperbolic secant, the inverse of Sech.
Cos -- The cosine of x, in radians.
Cosh -- Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$ .
Cot -- Cotangent, the reciprocal of Tan.
Csc -- Cosecant, the reciprocal of Sin.
Csgn -- The complex sign function -- $1$ in the right half-plane, $-1$ in the left, and on the imaginary axis the sign of the imaginary part; $0$ at the origin.
CubeRoot -- The real cube root of x.
DiracDelta -- The Dirac delta generalized function: 0 for any nonzero real argument, unevaluated at 0.
DiscreteDelta -- The Kronecker-delta-like discrete delta: 1 when every argument is exactly 0, 0 otherwise.
DiscreteShift -- The forward-shift operator: DiscreteShift(f, n) is f with every free n replaced by n + 1.
Exp -- The natural exponential function $e^x$ .
FractionalPart -- The fractional part of x, with the sign of x: x minus its integer part.
FunctionAnalytic -- True if f is analytic (entire) at every real x, False otherwise.
FunctionContinuouskernel -- Whether f is continuous for every real x (or, given a domain restriction, throughout it).
FunctionConvexity -- 1 if f is convex over its whole domain, -1 if concave, 0 if affine, Indeterminate otherwise.
FunctionDiscontinuities -- Where f fails to be continuous, as a condition in x.
FunctionDomain -- The set of real x where f (an expression in the single real variable x) is defined, as a condition in x.
FunctionInjective -- True if distinct x on f's domain always give distinct f(x), False otherwise.
FunctionMeromorphic -- True if f is meromorphic (a ratio of entire functions, poles allowed) everywhere, False otherwise.
FunctionMonotonicity -- 1 if f is increasing over its whole domain, -1 if decreasing, 0 if constant, Indeterminate otherwise.
FunctionPeriod -- The fundamental period of f in x, or 0 if f is provably not periodic.
FunctionRange -- The set of values f takes as x ranges over its domain, as a condition in y.
FunctionSign -- 1 if f is nonnegative over its whole domain, -1 if nonpositive, Indeterminate otherwise.
FunctionSingularities -- Where f fails to be analytic, as a condition in x.
FunctionSurjective -- True if every real y is f(x) for some x on f's domain, False otherwise.
Gudermannian -- The Gudermannian function $\operatorname{gd}(x) = 2\arctan(\tanh(x/2)) = \arctan(\sinh(x))$ , linking circular and hyperbolic functions without complex numbers.
HeavisideLambda -- The Heaviside-family triangle function: 1 - |x| for -1 <= x <= 1, 0 otherwise.
HeavisidePi -- The Heaviside-family rectangular pulse: 1 for |x| < 1/2, unevaluated at x = ±1/2, 0 for |x| > 1/2.
HeavisideTheta -- The Heaviside step function: 0 for x < 0, 1 for x > 0, unevaluated at x = 0.
IntegerPart -- The integer part of x: x truncated toward 0.
Lb -- The binary logarithm of z, identical to Log2.
Ln -- The natural logarithm, the inverse of Exp.
Log -- The logarithm of z in base b, defaulting to base 10.
Log10 -- The base-10 (common) logarithm of z.
Log2 -- The base-2 (binary) logarithm of z.
Mandelbrot -- The $n$ -th iterate of $z \mapsto z^2 + c$ starting from $0$ -- the complex value itself, not an escape count, so the same machinery that draws a zeta function draws this.
Ramp -- The ramp function: x for x >= 0, 0 for x < 0.
RealAbs -- The absolute value of a real number x, as a function defined only on the reals.
RealSign -- The sign of a real number x: -1, 0 or 1, as a function defined only on the reals.
Rescale -- Linearly rescale a value or list from one range to another.
SawtoothWave -- The sawtooth wave: x - floor(x), period 1, range [0, 1).
Sec -- Secant, the reciprocal of Cos.
Sin -- The sine of x, in radians.
Sinh -- Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$ .
SquareWave -- The square wave: period 1, default range {-1, 1}, value 1 on [0, 1/2) and -1 on [1/2, 1) of each period.
Tan -- The tangent of x, in radians.
Tanh -- Hyperbolic tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$ .
TriangleWave -- The triangle wave: period 1, default range {-1, 1}, sine-like phase (0 at x = 0, peak at x = 1/4).
UnitBox -- The unit box function: 1 on the closed [-1/2, 1/2], 0 outside.
UnitStep -- The unit step function: 0 for x < 0 and 1 for x >= 0.
UnitTriangle -- The unit triangle function: 1 - |x| for -1 <= x <= 1, 0 otherwise.

Geometric algebra

CliffordConjugate -- Clifford conjugation $\bar{x}$ : reversion after grade involution, costing sign $(-1)^{k(k+1)/2}$ per grade $k$ .
Dual -- The Poincaré dual $!x$ in algebra: each blade of $x$ replaced by the blade on the generators it does not contain, signed so $b \wedge {!b} = I$ .
Grade -- The single grade of a homogeneous multivector $x$ ; undefined if $x$ mixes grades.
GradeInvolution -- The grade involution $\hat{x}$ : negate every generator, costing sign $(-1)^k$ per grade $k$ .
GradePart -- The projection of a multivector $x$ onto grade $k$ .
LeftContraction -- The left contraction $a \lrcorner b$ : the geometric product of $a$ and $b$ at grade $|b|-|a|$ , zero when $a$ has the higher grade.
Pseudoscalar -- The top blade of an algebra — every generator, in canonical order, with coefficient one.
Reversion -- Reversion $\tilde{x}$ : reverse the order of generators in every blade of $x$ , costing sign $(-1)^{k(k-1)/2}$ per grade $k$ .
RightContraction -- The right contraction $a \llcorner b$ : the geometric product of $a$ and $b$ at grade $|a|-|b|$ , the mirror of LeftContraction.
Sandwich -- The sandwich product $a\,b\,\bar a$ : how a versor $a$ acts on $x$ , using the CliffordConjugate as the trailing factor.
ScalarProduct -- The scalar product of $a$ and $b$ — grade $0$ of their geometric product.

Graph statistics

AlbertsonIndex -- The sum over the edges of the difference of the degrees of its ends.
BridgeCount -- The edges whose deletion leaves more components.
ChromaticNumber -- The fewest colors for the vertices so that adjacent vertices differ.
CliqueNumber -- The most vertices that are pairwise adjacent.
ComplementComponentCount -- The connected components of the complement graph.
ConnectedInducedSubgraphCount -- The nonempty sets of vertices that induce a connected subgraph.
CutVertexCount -- The vertices whose deletion leaves more components.
CyclomaticNumber -- The edges less the vertices plus the components: the independent cycles.
DegreeTwoVertexCount -- The vertices with exactly two edges.
DistanceFourPairCount -- The pairs of vertices at distance 4.
DistanceThreePairCount -- The pairs of vertices at distance 3.
DistanceTwoPairCount -- The pairs of vertices at distance 2.
EvenDegreeVertexCount -- The vertices with an even number of edges.
FirstZagrebIndex -- The sum of the squares of the degrees.
ForgottenIndex -- The sum of the cubes of the degrees.
FourCliqueCount -- The sets of four vertices that are pairwise adjacent.
GraphComponentCount -- The connected components: the vertices that are the first of their component.
GraphEdgeCount -- The edges of the graph.
GraphLeafCount -- The vertices with exactly one edge.
HosoyaIndex -- The sets of edges with no two sharing a vertex, the empty set included.
IndependenceNumber -- The most vertices that are pairwise non-adjacent.
IndependentSetCount -- The sets of vertices, the empty set included, with no two adjacent.
InducedPathOnThreeVerticesCount -- The sets of three vertices that induce a path: exactly two of the three edges.
IsolatedVertexCount -- The vertices with no edge.
MatchingNumber -- The most edges with no two sharing a vertex.
MaximalIndependentSetCount -- The independent sets of vertices that no vertex can be added to.
MaximumDegree -- The greatest number of edges at one vertex.
MaximumDegreeVertexCount -- The vertices whose degree is the greatest.
MaximumDistance -- The greatest distance between two vertices: the diameter.
MinimumDegree -- The fewest edges at one vertex.
MinimumEccentricity -- The least eccentricity of a vertex, the greatest distance from it to another vertex: the radius.
OddDegreeVertexCount -- The vertices with an odd number of edges.
OreDegree -- The greatest sum of the degrees of the two ends of an edge.
SecondZagrebIndex -- The sum over the edges of the product of the degrees of its ends.
TotalIrregularity -- The sum over the unordered pairs of vertices of the difference of their degrees.
TriangleCount -- The triples of vertices that are pairwise adjacent.
TriangleVertexCount -- The vertices on a triangle.
VertexCoverNumber -- The fewest vertices that meet every edge: the vertices less the independence number.
WienerIndex -- The sum of the distances over the pairs of vertices that are joined by a path.

Graphics

Animator -- Wolfram's own Animator, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ArrayPlot -- Wolfram's own ArrayPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Arrow -- Wolfram's own Arrow, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
BarChart -- Wolfram's own BarChart, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
BarChart3D -- Wolfram's own BarChart3D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
BoxWhiskerChart -- Wolfram's own BoxWhiskerChart, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Chart -- The family chart head, held inert and drawn as whichever chart the data (or an explicit second argument naming the kind, e.g. "pie") calls for.
Checkbox -- Wolfram's own Checkbox, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Circle -- Wolfram's own Circle, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
CollectionTable -- Held inert and drawn as a table of a collection's elements.
ColorSlider -- Wolfram's own ColorSlider, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Column -- Wolfram's own Column, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ComplexPlot -- Wolfram's own ComplexPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ComplexPlot3D -- Wolfram's own ComplexPlot3D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ContourPlot -- Wolfram's own ContourPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Dendrogram -- Wolfram's own Dendrogram, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
DensityPlot -- Wolfram's own DensityPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
DiscretePlot -- Wolfram's own DiscretePlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Disk -- Wolfram's own Disk, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Dynamic -- Wolfram's own Dynamic, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
DynamicModule -- Wolfram's own DynamicModule, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
GraphPlot -- Wolfram's own GraphPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Graphics3D -- Wolfram's own Graphics3D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Grid -- Wolfram's own Grid, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Histogram -- Given data alone, held inert and drawn as a histogram picture; given a bin spec too, computed as compute-engine's own $\mathrm{Histogram}$ does.
Image -- A picture as a value, carrying a data: or other URI rather than pixels, so it serializes the same way in Node and in a browser; see Rasterize to build one from an SVG document.
InputField -- Wolfram's own InputField, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
IntervalSlider -- Wolfram's own IntervalSlider, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Knob -- Held inert and drawn as a rotary control bound to var over range, a $(\mathrm{min}, \mathrm{max})$ or $(\mathrm{min}, \mathrm{max}, \mathrm{step})$ tuple.
Labeled -- Wolfram's own Labeled, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
LayeredGraphPlot -- Wolfram's own LayeredGraphPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Line -- Wolfram's own Line, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListContourPlot -- Wolfram's own ListContourPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListDensityPlot -- Wolfram's own ListDensityPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListLinePlot -- Wolfram's own ListLinePlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListPicker -- Wolfram's own ListPicker, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListPlot -- Wolfram's own ListPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListPlot3D -- Wolfram's own ListPlot3D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ListPolarPlot -- Wolfram's own ListPolarPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Locator -- Wolfram's own Locator, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Manipulate -- Wolfram's own Manipulate, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
PaneSelector -- Wolfram's own PaneSelector, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Panel -- Wolfram's own Panel, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
ParametricPlot -- Wolfram's own ParametricPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
PieChart -- Wolfram's own PieChart, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Plot -- Wolfram's own Plot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Plot3D -- Wolfram's own Plot3D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Point -- Wolfram's own Point, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
PolarPlot -- Wolfram's own PolarPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
PopupMenu -- Wolfram's own PopupMenu, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
RadioButtonBar -- Wolfram's own RadioButtonBar, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Rectangle -- Wolfram's own Rectangle, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Row -- Wolfram's own Row, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
SetterBar -- Wolfram's own SetterBar, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Show -- Wolfram's Show: layers drawn in one coordinate frame, sharing one view. Held as written, so a layer's ColorRules name the properties it publishes rather than compute-engine's heads.
Slider -- Wolfram's own Slider, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Slider2D -- Wolfram's own Slider2D, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
StreamPlot -- Wolfram's own StreamPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Toggler -- Wolfram's own Toggler, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
TogglerBar -- Wolfram's own TogglerBar, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Tooltip -- Wolfram's own Tooltip, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
TorusSquare -- The torus as a square with its opposite edges glued, and the torus knot T(p, q) on it as a straight line of slope q/p; held inert and drawn as a diagram GraphicsBox.
TreeGraph -- Wolfram's own TreeGraph, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
Variable -- A declared variable's domain, starting value and hints, in a Variables option: the domain picks the control a hole showing it gets. Held as written.
VectorPlot -- Wolfram's own VectorPlot, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).
VerticalSlider -- Wolfram's own VerticalSlider, held inert -- the expression is what a worksheet or REPL draws, not something this library computes (see @enumeratio/formats/src/graphics.ts).

Group algebras

ClassSum -- The $k$ -th class sum: add up one conjugacy class. Class sums are the basis of the CENTRE of $k[G]$ — a commutative subalgebra of a usually non-commutative algebra.
ConjugacyClasses -- The conjugacy classes of a finite group — the orbits of $g \mapsto xgx^{-1}$ . Their number is the dimension of the centre of $k[G]$ .
GroupBasis -- A basis element of the group algebra $k[G]$ , named by its group element's label. The product is the group's own multiplication, extended bilinearly.

Hecke algebras

HeckeAlgebra -- The Iwahori–Hecke algebra $H_n(q)$ , the $q$ -deformation of the group algebra of $S_n$ , with basis HeckeT.
HeckeIdentity -- The identity element $T_e$ of $H_n(q)$ , for writing products that start from one.
HeckeSpecialize -- Substitute a value for the deformation parameter. At $q = 1$ the algebra collapses to the group algebra of $S_n$ — which is what makes 'deformation' the right word.
HeckeT -- The basis element $T_w$ of the Iwahori–Hecke algebra $H_n(q)$ , indexed by a permutation in one-line notation. Multiplication is the $q$ -deformation of the symmetric group's.

Hopf algebras

Antipode -- The antipode $S$ , the last piece of a Hopf algebra: the unique map with $m(S \otimes \mathrm{id})\Delta = \eta\varepsilon$ .
ConjugateComposition -- The conjugate (transpose) composition $\alpha^{*}$ : the transpose of the ribbon's skew shape, gotten by complementing the descent set inside $\{1,\ldots,n-1\}$ and reversing.
Coproduct -- The coproduct $\Delta$ : takes one element to a sum of HopfTensor pairs. Deconcatenation for QSym; the multiplicative extension of splitting a part for NSym.
Counit -- The counit $\varepsilon$ : picks out the coefficient of the empty composition, the degree-0 part of element.
HopfDegree -- The degree of a homogeneous element — the sum of its composition's parts; undefined when element mixes degrees.
HopfTensor -- One tensor pair of a Coproduct expansion, $\text{left} \otimes \text{right}$ — its own head because CircleTimes already names the shared ordered product.
InCompleteBasis -- Rewrite an element of NSym in the complete basis $H_\alpha$ .
InFundamentalBasis -- Rewrite an element of QSym in the fundamental basis QSymF.
InMonomialBasis -- Rewrite an element of QSym in the monomial basis QSymM.
InRibbonBasis -- Rewrite an element of NSym in the ribbon basis NSymR.
NSymAlgebra -- The degree- $n$ graded piece of NSym — one basis element per composition of $n$ , $2^{n-1}$ of them for $n \geq 1$ .
NSymH -- The complete basis $H_\alpha$ of NSym, whose product is concatenation of compositions.
QSymAlgebra -- The degree- $n$ graded piece of QSym — one basis element per composition of $n$ , $2^{n-1}$ of them for $n \geq 1$ .
QSymF -- The fundamental basis $F_\alpha$ of QSym, dual to the ribbon basis NSymR.
QSymM -- The monomial basis $M_\alpha$ of the quasi-symmetric functions, indexed by a composition. The product is the quasi-shuffle, which may ADD two leading parts as well as interleave them.

Hypercomplex algebra

AlgebraSignature -- The signature vector: what each of the algebra's generators squares to, in order — $-1$ , $+1$ or $0$ .
DualAlgebra -- The multi-dual algebra $\mathbb{R}[\varepsilon_1,\ldots,\varepsilon_n]/(\varepsilon_k^2)$ on $n$ commuting nilpotent generators, each squaring to $0$ .
Expand -- Expand out products and positive integer powers; for a hypercomplex operand, puts $x$ in blade normal form — the same result the multivector arithmetic already produces.
MulticomplexAlgebra -- The multicomplex algebra $\mathbb{C}_n = \mathbb{R}[i_1,\ldots,i_n]/(i_k^2+1)$ on $n$ commuting generators, each squaring to $-1$ .
Norm -- The algebra norm of a hypercomplex element: the determinant of multiplication-by- $z$ on the $2^n$ -dimensional space. Reduces to the Gaussian $a^2+b^2$ at one imaginary unit.
Quaternions -- $\mathbb{H}$ , the quaternions — which IS $\mathrm{Cl}(0,2)$ : two anticommuting generators squaring to $-1$ , with $k = f_1f_2$ as the third unit.
SplitAlgebra -- The multi-perplex algebra $\mathbb{R}[j_1,\ldots,j_n]/(j_k^2-1)$ on $n$ commuting generators, each squaring to $+1$ .

Incidence algebras

BooleanLattice -- The poset of subsets of $\{1,\dots,n\}$ ordered by inclusion, on which MoebiusFunction gives the signs of inclusion–exclusion.
Chain -- The poset $1 < 2 < \cdots < n$ , on which MoebiusFunction is $1$ on a point, $-1$ on a cover, and $0$ on anything longer.
DivisorLattice -- The poset of divisors of $n$ ordered by divisibility, on which MoebiusFunction recovers the classical number-theoretic $\mu$ .
IncidenceAlgebra -- The incidence algebra of a finite poset — basis PosetInterval, product composition of intervals, invertible zeta function whose inverse is MoebiusFunction.
MoebiusFunction -- The Möbius function $\mu(x,y)$ of a finite poset — the inverse of the zeta function in the incidence algebra. Specialising the poset recovers number theory's $\mu$ and inclusion–exclusion.
MoebiusInvert -- Möbius inversion: given $g(y) = \sum_{x \le y} f(x)$ , recover $f$ . The inverse of PosetSumDown, and the reason the incidence algebra is worth having.
PosetElements -- The elements of a finite poset, in a linear extension — the order every other head here indexes values by.
PosetInterval -- A basis element of the incidence algebra — the interval $[x,y]$ of the poset, for $x \le y$ .
PosetSumDown -- Sum a function down the poset's order: $g(y) = \sum_{x \le y} f(x)$ , the map MoebiusInvert undoes.
PosetZeta -- The zeta function $\zeta(x,y)$ of a finite poset — $1$ when $x \le y$ and $0$ otherwise — whose inverse in the incidence algebra is MoebiusFunction.

Interval arithmetic

Around -- A number x with an uncertainty dx, propagated to first order through arithmetic and functions -- the same uncertainty-arithmetic layer as Interval and CenteredInterval. Provided by @enumeratio/analytic, though the head's natural home is @enumeratio/statistics.
CenteredInterval -- A ball c ± r with center-radius arithmetic -- the complex-capable sibling of Interval. Provided by @enumeratio/analytic.
Interval -- Arithmetic over compute-engine's native Interval(a, b) -- extended in place by @enumeratio/analytic, since compute-engine declares it only as a real set with no arithmetic of its own. A result contains every value the operation takes on its inputs.

Linear algebra

MatrixExp -- The matrix exponential $e^A = \sum_{k \ge 0} A^k / k!$ , for a square matrix A.
MatrixFunction -- A scalar function f extended to a square matrix m, via its eigendecomposition. Provided by @enumeratio/analytic.
MatrixRank -- The rank of a matrix: the number of linearly independent rows (equivalently, columns).
Rank -- The number of dimensions of an array: 0 for a scalar, 1 for a vector, 2 for a matrix.

Modular arithmetic

AdditionTable -- The addition table of ℤ/n, R = QuotientRing(Integers, n): the matrix whose (i, j) entry is aᵢ + aⱼ, its rows and columns ℤ/n's elements in the order ElementOrder lists them.
ChineseRemainder -- The smallest non-negative integer congruent to each ri modulo the corresponding mi.
IntegerMod -- $k \bmod n$ as a VALUE, after Sage's Mod(k, n): ResidueClass whose classes of different moduli combine in $\mathbb{Z}/\gcd(m, n)$ .
IntegerModRing -- The old spelling of QuotientRing(Integers, m), kept working: IntegerModRing(m) evaluates to its canonical form.
Mod -- The remainder of a on division by b.
ModularInverse -- The $x$ with $a x \equiv 1 \pmod m$ , when $a$ is a unit mod $m$ .
MultiplicationTable -- The multiplication table of ℤ/n, R = QuotientRing(Integers, n): the matrix whose (i, j) entry is aᵢaⱼ, its rows and columns ℤ/n's elements in the order ElementOrder lists them.
MultiplicativeOrder -- The smallest positive k such that a^k ≡ 1 (mod n).
PowerMod -- Modular exponentiation: a^b mod m, computed without forming a^b directly.
PowerModList -- Every $x$ in $[0, m)$ with $x^r \equiv a^s \pmod m$ — all the values $a^{s/r}$ can take modulo $m$ .
PrimitiveRootList -- Every primitive root of $n$ — every generator of $(\mathbb{Z}/n)^\times$ — ascending.
QuotientRing -- The ring $\mathbb{Z}/m\mathbb{Z}$ : compute-engine's QuotientRing, made the finite collection of its $m$ ResidueClass classes.
ResidueClass -- $k \bmod n$ as a VALUE — an element of $\mathbb{Z}/n\mathbb{Z}$ that arithmetic stays inside, after Sage's Mod(a, m). compute-engine's head, strict about moduli: IntegerMod is its Sage-flavoured sibling.

Number theory 8 primitive

Abundance -- How far the divisors of n overshoot twice n: $\sigma(n) - 2n$ .
AlgebraicIntegerQkernel -- Whether the algebraic number $a$ is an algebraic integer: a root of a monic polynomial over $\mathbb{Z}$ .
AlgebraicIntegers -- The ring of integers $\mathcal{O}_K$ of the number field $K = \mathbb{Q}(a)$ : every element of $K$ that is a root of a monic polynomial over $\mathbb{Z}$ , its maximal order.
AlgebraicNumberNormkernel -- The norm of $a$ over $\mathbb{Q}$ in the field $\mathbb{Q}(a)$ : the product of its conjugates.
AlgebraicNumberTracekernel -- The trace of $a$ over $\mathbb{Q}$ in the field $\mathbb{Q}(a)$ : the sum of its conjugates.
AlgebraicOrder -- The order $\mathbb{Z}[a]$ generated by the algebraic integer $a$ : the polynomials in $a$ over $\mathbb{Z}$ , a subring of finite index in the ring of integers of $\mathbb{Q}(a)$ .
AliquotSum -- The sum of the proper divisors of n: $\sigma(n) - n$ .
CarmichaelLambda -- The Carmichael function $\lambda(n)$ : the exponent of the multiplicative group mod $n$ — the least $m$ with $a^m\equiv1\pmod n$ for every $a$ coprime to $n$ .
CongruentMod -- Whether $a \equiv b \pmod m$ -- true exactly when $m \mid (a - b)$ .
DivisorSigma -- The sum of the kth powers of the divisors of n.
DivisorSum -- The sum of $f(d)$ over the positive divisors $d$ of $n$ , optionally only those where $\mathrm{cond}(d)$ holds.
Divisors -- All positive divisors of n, in increasing order.
EulerE -- The Euler numbers $E_n$ and Euler polynomials $E_n(x)$ .
ExtendedGCD -- The GCD of the arguments together with Bézout coefficients x₁, x₂, … such that a₁·x₁ + a₂·x₂ + … = GCD(a₁, a₂, …).
FactorInteger -- The prime factorisation of n as a list of [prime, exponent] pairs.
FrobeniusNumber -- The largest integer that is not a non-negative integer combination of the given positive integers.
FrobeniusSolve -- All non-negative integer solutions $x$ of $a\cdot x = b$ , lexicographically.
GCD -- The greatest common divisor of the arguments.
HermiteDecomposition -- $\{u, h\}$ with $u$ unimodular and $u \cdot m = h$ upper triangular in Hermite normal form.
IntegerExponent -- The largest $k$ with $b^k \mid n$ — the $b$ -adic valuation of $n$ as an integer.
IsCoprime -- Tests whether the arguments are pairwise relatively prime.
IsPerfect -- Tests whether $n$ equals the sum of its proper divisors.
IsPrime -- Tests whether n is a prime number.
IsPrimePower -- Tests whether $n$ is a positive integer power of a single prime.
IsSquareFree -- Tests whether n has no repeated prime factors.
JacobiSymbol -- The Jacobi symbol (n/m), generalising the Legendre symbol to composite m.
KroneckerSymbol -- The Kronecker symbol (a/n): the Jacobi symbol extended to every integer n.
LCM -- The least common multiple of the arguments.
LegendreSymbol -- The Legendre symbol (n/p): whether n is a quadratic residue mod the odd prime p.
LiouvilleLambda -- The Liouville function $\lambda(n)=(-1)^{\Omega(n)}$ .
MangoldtLambda -- The von Mangoldt function: $\ln p$ if $n$ is a power of the prime $p$ , else 0.
MersennePrimeExponent -- The exponent $p$ of the $n$ th Mersenne prime $2^p-1$ .
MinimalPolynomialkernel -- The minimal polynomial of the algebraic number $a$ in $x$ : the irreducible polynomial over $\mathbb{Z}$ with positive leading coefficient that $a$ is a root of.
MoebiusMu -- The Möbius function: 0 if n has a squared prime factor, else (-1)^(number of prime factors).
NextPrime -- The next prime strictly greater than n, or the kth prime after n.
NthPrime -- The nth prime number (NthPrime(1) = 2).
NumberFieldDiscriminantkernel -- The discriminant of the number field $\mathbb{Q}(a)$ : of its ring of integers $\mathcal{O}_K$ .
NumberFieldIntegralBasiskernel -- A $\mathbb{Z}$ -basis of the ring of integers of $\mathbb{Q}(a)$ , written on $a$ 's powers as Wolfram writes it.
NumberFieldSignaturekernel -- The signature $(r_1, r_2)$ of $\mathbb{Q}(a)$ : its real embeddings, and its pairs of complex conjugate ones.
PartitionsQ -- The number $q(n)$ of partitions of $n$ into distinct parts.
PerfectNumber -- The $n$ th perfect number.
PolynomialRootkernel -- The $k$ -th root of the polynomial $p$ in its one unknown, in Wolfram's order: real roots first, increasing, then complex roots by real part and imaginary part. Wolfram's Root; compute-engine's Root is the $n$ -th root.
PowersRepresentations -- The ways to write $n$ as a sum of $k$ non-negative $p$ th powers, as non-decreasing lists.
PrimeNu -- The number of distinct prime factors of n.
PrimeOmega -- The number of prime factors of n, counted with multiplicity.
PrimePi -- The prime-counting function: how many primes are ≤ n.
QuadraticIntegers -- The ring of integers of $\mathbb{Q}(\sqrt d)$ : $\mathbb{Z}[\omega]$ with $\omega = (-1 + \sqrt d)/2$ when $d \equiv 1 \pmod 4$ , else $\sqrt d$ .
QuadraticOrder -- The quadratic order of discriminant $D = f^2 D_K$ : $\mathbb{Z} + f\,\mathcal{O}_K$ , the subring of conductor $f$ in the ring of integers of $\mathbb{Q}(\sqrt D)$ . $\mathbb{Z}[\sqrt{d}]$ is $\mathrm{QuadraticOrder}(4d)$ .
Quotient -- The integer quotient of $m$ by $n$ — for Gaussian integers, $m/n$ rounded to the nearest lattice point.
RamanujanTau -- Ramanujan's tau function: the coefficients of the discriminant modular form $\Delta(q) = q\prod_{k\ge1}(1-q^k)^{24}$ .
RationalReconstruction -- The small fraction $n/d$ whose image in $\mathbb{Z}/m$ is $a$ — the inverse of reading $n/d$ as $n \cdot d^{-1} \bmod m$ .
SquareFreePart -- The square-free part $q$ of $n=\pm s^2 q$ : the product of the primes dividing $n$ to an odd power.
SquarePartRoot -- The root $s$ of the square part of $n=\pm s^2 q$ : the largest $s$ with $s^2$ dividing $n$ .
SquaresR -- The number of ways to write $n$ as a sum of $d$ squares, counting signs and order.
Totient -- Euler's totient function: the count of integers in 1..n coprime to n.
XGCD -- The extended Euclidean algorithm -- $(g, x, y)$ with $g = \gcd(a, b) = ax + by$ .

Numeral systems

AdicDigits -- The digits of a AdicNumeral as a list, least significant first — the only order that lists something with no left end.
AdicExpansion -- The digits of a $b$ -adic number, written with the infinite end on the left: $\ldots 6667$ for $1/3$ in $\mathbb{Z}_{10}$ , $0.12$ for $7/25$ in $\mathbb{Q}_5$ , $+ O(b^n)$ when capped.
AdicNumeral -- A $b$ -adic number: the rational $x$ read in $\mathbb{Z}_b$ (or $\mathbb{Q}_p$ for prime $b$ ), exact when built from a rational, $+ O(b^{prec})$ when capped. The ring operations work on it.
AdicNumerals -- The first prec $b$ -adic digits, always fixed-width: the sign is folded into the digits rather than a sign bit, giving a bijection between $(-b^{prec}/2, b^{prec}/2]$ and width-prec digit strings.
AdicSqrt -- A square root of a AdicNumeral in $\mathbb{Q}_p$ , found by Hensel lifting — defined only for prime bases, and only when one exists.
AdicUnitPart -- The unit part $u$ of a AdicNumeral $x = b^{v_b(x)} \cdot u$ .
AdicValuation -- The $b$ -adic valuation $v_b(x)$ — the power of $b$ dividing $x$ — with its companions: the norm $|x|_b = b^{-v}$ and the unit part $u$ in $x = b^v · u$ .
BalancedNumerals -- Balanced base $b$ (odd): digits run $-(b-1)/2,\dots,(b-1)/2$ — it spells negative integers with no sign at all, and rounding to nearest is just truncation. Balanced ternary ( $b=3$ ) is the famous case.
BalancedRadix -- The old spelling of BalancedNumerals, kept working: BalancedRadix(b) evaluates to its canonical form.
BijectiveNumerals -- Bijective base $k$ : digits $1,\dots,k$ , with no zero digit at all — every positive integer has exactly one representation, and zero is the empty numeral. Bijective base 26 is how spreadsheet columns are lettered (A, …, Z, AA, …).
BijectiveRadix -- The old spelling of BijectiveNumerals, kept working: BijectiveRadix(k) evaluates to its canonical form.
CombinatorialNumerals -- The combinatorial number system of degree $k$ : $n \leftrightarrow$ the strictly decreasing tuple $c_k > \dots > c_1 \ge 0$ with $n = \binom{c_k}{k} + \dots + \binom{c_1}{1}$ — the unranking of $k$ -subsets in colexicographic order.
CombinatorialSystem -- The old spelling of CombinatorialNumerals, kept working: CombinatorialSystem(k) evaluates to its canonical form.
DigitCount -- Counts of each digit (1 through 9, then 0) occurring in n, base 10 by default.
DigitSum -- The sum of the digits of n in the given base (default 10).
FactorialNumerals -- Factoradic: place $k$ has weight $k!$ and digit at most $k$ . The digits of $n$ are the Lehmer code of the $n$ -th permutation in lexicographic order, so unranking a permutation and writing a number in this system are the same operation.
FromDigits -- The integer a digit string denotes, in any numeral system. The inverse of IntegerDigits — and the round trip is the whole specification of a system.
HenselLift -- The $b$ -adic root of a polynomial that reduces to seed mod $b$ , by Newton's iteration — Hensel's lemma made to run. AdicSqrt is the special case $f = x^2 − a$ .
IntegerDigits -- The digits of $n$ , most significant first — base 10 by default. The base slot takes a whole numeral SYSTEM, not only an integer — factorial, Zeckendorf, Ostrowski, balanced, negative, bijective, mixed-radix, primorial, combinatorial or residue.
IntegerLength -- The number of digits of $n$ in the given base — base 10 by default.
IntegerReverse -- The integer whose digits are those of $n$ in reverse order.
IntegerString -- The string representation of n in the given base (default 10).
NegativeNumerals -- Base $-b$ : digits $0,\dots,b-1$ again, but the place values alternate sign, so every integer — negatives included — has exactly one representation and there is no sign bit.
NegativeRadix -- The old spelling of NegativeNumerals, kept working: NegativeRadix(b) evaluates to its canonical form.
NumberExpand -- The place-value terms of $n$ : each digit times its power of the base.
NumeralSystemShape -- What a system's numerals look like, as a Dictionary: whether it is a bijection, which integers it spells, and with which digits.
Ostrowski -- The old spelling of OstrowskiNumerals, kept working: Ostrowski(quotients) evaluates to its canonical form.
OstrowskiNumerals -- Ostrowski numeration: the numeral system a continued fraction's partial quotients define, with a ceiling rule — a digit may reach its bound only if the one below it is zero — that forbids carries. All quotients equal to 1 gives the golden ratio and reduces to ZeckendorfNumerals.
PositionalNumerals -- Ordinary base $b$ notation, non-negative integers only — the same digits IntegerDigits already gives for an integer base, wrapped as a numeral-system value so it can stand wherever the others do.
PrimorialNumerals -- Primorial base: place $k$ has weight $p_1 \cdots p_k$ and digit below $p_{k+1}$ — the same idea as FactorialNumerals with the primes in place of the naturals.
Radix -- The old spelling of PositionalNumerals, kept working: Radix(b) evaluates to its canonical form.
RealDigits -- The digits of a real number and the position of its decimal point, with the repeating block of a rational marked as a sublist.
ResidueNumerals -- A residue number system: the digits are $n \bmod$ each modulus, with no place values at all — which is what makes addition and multiplication carry-free and parallel. It is a bijection onto $[0, \prod m_i)$ exactly when the moduli are pairwise coprime.
ResidueSystem -- The old spelling of ResidueNumerals, kept working: ResidueSystem(moduli) evaluates to its canonical form.
RomanNumeral -- The Roman numeral for a non-negative integer, as a string.
ZeckendorfNumerals -- Zeckendorf: binary digits over the Fibonacci place values $1,2,3,5,8,\dots$ — every positive integer has exactly one representation with no two adjacent ones, and greedy digit choice finds it.

Parking function statistics

CenterSize -- The length of the longest subsequence a(1), ..., a(k) with a(i) <= i.
CriticalLeftToRightMaxima -- The entries p that are left to right maxima with exactly p - 1 entries smaller and n - p larger.
Deficiencies -- Positions i with p(i) < i.
InitialIncreasingSegmentLength -- The length of the initial strictly increasing segment.
LuckyCars -- The cars that park in the spot they prefer.
MaximalParkingDisplacement -- The most spots any car drives past its preferred spot.
SumLessLength -- The sum of the preferences less their number.
Ties -- Positions i with p(i) = p(i+1).
TotalParkingDisplacement -- The sum, over the cars, of the spots they drive past: the sum of i - p(i).

Partition and standard tableau statistics

WeightedSize -- The sum over the cells of their row, counting rows from 0.

Partition statistics

AddableCells -- Cells that can be added to the diagram and leave a partition: one more than the distinct parts.
Aft -- The size minus the larger of the largest part and the number of parts.
AlladiWeight -- The last part times the product over consecutive parts of the gap less 1.
ArmOfFirstCell -- The arm of cell (1,1): the first part minus one.
BerkovichGarvanRank -- The sum of a checkerboard filling of the diagram, +1 on cells whose coordinates sum to an even number and -1 on the others.
BorderSkewStandardTableauCount -- Standard tableaux of the skew shape that adds a cell to every column of the diagram: the size factorial over the factorials of the cells added to each row (the size less the largest part, then the gaps between consecutive parts).
BraceletCount -- The colorings of a polygon with these color multiplicities, up to rotation and reflection: the necklaces and the colorings fixed by the reflections, over twice the size.
BulgarianSolitaireOrbitSize -- The partitions that Bulgarian solitaire (remove the first column, insert it as a row) visits from this one, itself included.
BulgarianSolitairePreimageCount -- The partitions of the same size that Bulgarian solitaire (remove the first column, insert it as a row) sends to this one: the distinct part sizes of at least the number of parts less one.
BulgarianSolitairePreperiod -- The steps of Bulgarian solitaire before this partition reaches the cycle it ends in.
CartanDeterminant -- The determinant of the matrix of the minimums of the distinct parts: the least distinct part times the gaps between consecutive distinct parts.
CellsOutsideDurfeeSquare -- Cells of the diagram outside its Durfee square.
CellsOutsideFirstHook -- Cells in neither the first row nor the first column: the size minus the hook length of cell (1,1).
CellsOutsideFirstRow -- Cells of the diagram below the first row: the size minus the largest part.
CellsOutsideLargestStaircase -- Cells of the diagram outside the largest staircase it contains.
CellsWithEqualArmAndLeg -- Cells whose arm and leg are equal.
CellsWithZeroLegAndOddArm -- Cells with leg 0 and an odd arm.
CentralizerSize -- The size of the centralizer of a permutation with these cycle lengths: the product of the parts times the product of the factorials of the multiplicities.
CoarseningCount -- The partitions that coarsen this one, whose parts subdivide to give its. Itself included.
CohookLengthProduct -- The product of the cohook lengths i + j - 1 of the cells.
CompleteHomogeneousSymmetricFunctionAtOne -- The complete homogeneous symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the product of the multiset coefficients of the number of parts and each part.
ConjugateDistinctParts -- Distinct part sizes of the conjugate.
ConjugateOddParts -- Odd parts of the conjugate — equivalently, the distinct part sizes of λ.
ContainingRectangleShortSide -- The shorter side of the smallest rectangle that contains the diagram: the smaller of the largest part and the number of parts.
ContainingSquareSide -- The side of the smallest square that contains the diagram: the larger of the largest part and the number of parts.
ContentSum -- The sum of the contents j - i of the cells.
CornerMajorIndex -- The sum, over the corners, of the step number along the boundary path from the bottom left: (number of parts - i) + the i-th part.
Corners -- Corner cells — parts strictly larger than the next part (the last part always counts).
CouplesAreForeverGrundyValue -- The Grundy value of the game that splits a part above 2 into two parts: the nim-sum of the values of the parts, each the least value missing from the nim-sums over its splits.
Crank -- The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
CubeRootCount -- The permutations whose cube is a fixed permutation of this cycle type: the product over part sizes d, with a parts of size d, of the number of ways to take the cube root of each part of size d alone or of three together as a cycle of length 3d.
CubedStandardTableauCount -- The cube of the number of standard tableaux.
CyclicSievingPolynomialDegree -- (1 - 1/largest part) times the least common multiple of the parts: the degree of the cyclic sieving polynomial of a permutation with these cycle lengths.
DiagonalArmSum -- The sum of the arms of the cells on the main diagonal.
DiagonalHookLengthProduct -- The product of the hook lengths of the cells on the main diagonal.
DiagonalInversionDefect -- Cells whose arm minus leg is neither 0 nor 1.
DiagonalInversions -- Cells whose arm minus leg is 0 or 1.
DistinctEvenParts -- Distinct even part sizes.
DistinctEvenSubsetSums -- Distinct nonzero even sums of sub-multisets of the parts.
DistinctMultiplicities -- How many different multiplicities the part sizes have.
DistinctOddPartsBelowLargestEvenPart -- Distinct odd part sizes smaller than the largest even part.
DistinctOddSubsetSums -- Distinct odd sums of sub-multisets of the parts.
DistinctParts -- How many distinct part sizes occur.
DistinctPartsEqualToMultiplicity -- Distinct part sizes equal to their multiplicity.
DistinctRepeatedParts -- Distinct part sizes that occur at least twice.
DistinctSubsetSums -- Distinct sums of non-empty sub-multisets of the parts.
DominatedPartitionCount -- The partitions of the same size that this one dominates (partial sums of the parts at least those of the other), itself included.
DurfeeSquare -- The side of the Durfee square: the largest d with at least d parts of size at least d.
DysonRank -- Largest part minus number of parts.
EdgeOrbitCount -- Orbits of the complete graph's edges under a permutation with these cycle lengths: the sum of half the parts rounded down, plus the sum of gcd(part, part) over pairs.
ElementarySymmetricFunctionAtOne -- The elementary symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the product of the binomial coefficients of the number of parts and each part.
EvenOddCrank -- The largest even part minus the number of odd parts.
EvenParts -- Parts that are even.
ExponensConsonantiae -- The least common multiple of the parts over their greatest common divisor.
FirstRowHookLengthSum -- The sum of the hook lengths in the first row: the size plus the binomial of the largest part and 2.
FlooredHalfMultiplicitySum -- The sum, over part sizes, of half the multiplicity rounded down.
GiniIndex -- The size plus 1 choose 2, less the sum over the columns of the column length plus 1 choose 2.
GreatestCommonDivisorOfParts -- The greatest common divisor of the parts.
HalfLargestEvenPart -- Half the largest even part; 0 if there is none.
HalfPerimeter -- Half the perimeter of the bounding rectangle of the Young diagram: largest part plus number of parts.
HalfSumOfEvenParts -- Half the sum of the even parts.
HookLengthOfFirstCell -- The hook length of cell (1,1): the largest part plus the number of parts, less one.
HookLengthOfLastDiagonalCell -- The hook length of the last cell on the main diagonal: that of cell (d,d) for Durfee side d.
HookLengthSquareSum -- The sum of the squares of all hook lengths.
HookProduct -- The product of all hook lengths — the denominator in the hook-length formula.
HurwitzNumber -- Hurwitz's number for the partition: (n + c - 2)! n^(c-3) times the product of p^p / (p-1)! over its c parts p, n being its size.
InterlacingPartitionCount -- The number of partitions that interlace this one: the product over the parts of the gap to the next part (0 after the last) plus 1.
InvariantGraphCount -- The simple graphs on the points that a permutation of this cycle type maps to themselves: 2 to the number of edge orbits.
InvariantOrientedCycleCount -- The cyclic permutations that a permutation of this cycle type fixes under conjugation: none unless the parts are all equal to some d, in which case totient(d) d^(m-1) (m-1)! for m parts.
InvariantSubsetCount -- Subsets of the points left invariant by a permutation with these cycle lengths: each cycle is in or out, 2 to the number of cycles.
InvariantThreeElementSubsetCount -- Invariant 3-element subsets: three fixed points, a fixed point and a 2-cycle, or a 3-cycle.
InvariantTwoElementSubsetCount -- Invariant 2-element subsets: a pair of fixed points or a 2-cycle.
IsSelfConjugate -- 1 when λ equals its conjugate, 0 otherwise.
KrewerasNumber -- The number of noncrossing set partitions whose block sizes are the parts: the size factorial over the factorial of size + 1 - length and the factorials of the multiplicities.
LagrangeInversionCoefficient -- The coefficient of the monomial in the compositional-inverse series: n (n + 1) ... (n + length - 2) over the factorials of the multiplicities.
LargestEvenPart -- The largest even part; 0 if there is none.
LargestMissingPart -- The largest integer below the largest part that is not a part; 0 if every smaller integer is.
LargestMultiplicity -- The largest multiplicity of a part.
LargestPart -- The largest part.
LargestRepeatedPart -- The largest part that occurs at least twice; 0 if the parts are distinct.
LargestShiftedPart -- The largest part after adding i - 1 to the i-th part.
LargestStaircaseSide -- The side of the largest staircase (k, k-1, ..., 1) inside the diagram.
LeastCommonMultipleOfParts -- The least common multiple of the parts.
LegOfFirstCell -- The leg of cell (1,1): the number of parts minus one.
LengthCrank -- The number of parts when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
LogarithmicDerivativeCoefficient -- The coefficient of the monomial in the logarithmic derivative of 1 - c_1 x - c_2 x^2/2! - ..., times a factorial: the set partitions with these block sizes, times (length - 1)! for arranging the blocks in a cycle.
LowerCoverCountInDominanceOrder -- The partitions of the same size that this one covers in dominance order: the upper covers of its conjugate.
MaximalRectangleHalfPerimeter -- Half the perimeter of the rectangle inside the diagram that has the greatest perimeter.
MaximalRookPlacementCount -- The ways to place as many non-attacking rooks as possible on the Ferrers board of the diagram.
MinimalSubpartitionSizeDifference -- The smallest difference in size between the two halves of a split of the parts into two sub-multisets.
MonotoneHurwitzNumber -- The monotone factorizations of genus zero of a permutation of this cycle type into transpositions: the product of the Catalan numbers C(p - 1) over the parts p.
MultinomialOfParts -- The multinomial coefficient of the size over the parts: the size factorial over the product of the factorials of the parts.
MultipartitionCount -- Tuples of partitions whose sizes are the parts: the product of the partition numbers of the parts.
MultiplicityFactorialProduct -- The product of the factorials of the multiplicities of the parts.
MultiplicityOfLargestPart -- How many parts equal the largest.
MultiplicityOfSmallestPart -- How many parts equal the smallest.
NecklaceCount -- The colorings of a cycle with these color multiplicities, up to rotation: the size's divisors d of the greatest common divisor of the parts, each contributing totient(d) multinomials, over the size.
NonDominatingSubmultisetCount -- The distinct sub-multisets of the parts that, read as partitions, do not dominate their own conjugate.
OddParts -- Parts that are odd.
OddPartsBelowLargestEvenPart -- Odd parts smaller than the largest even part.
OrderedRefinementCount -- The partitions whose parts cut, in order, into consecutive blocks summing to the parts of this one.
PairsOfDistinctParts -- Pairs of parts that differ in size: the degree of the polynomial counting semistandard tableaux of the stretched shape.
PairwiseGreatestCommonDivisorSum -- The sum of the greatest common divisors of all pairs of parts.
PairwiseProductSum -- The sum of the products of all pairs of parts.
PartCount -- The number of parts.
PartFactorialProduct -- The product of the factorials of the parts.
PartProduct -- The product of the parts.
PartRange -- The largest part minus the smallest.
PartRearrangementCount -- Distinct rearrangements of the parts, the compositions that sort to this partition: the number of parts factorial over the factorials of the multiplicities.
PartSquareProduct -- The product of the squares of the parts.
PartSquareSum -- The sum of the squares of the parts.
PartitionSize -- The sum of the parts.
PartsAtLeastTwo -- Parts of size at least 2.
PartsEqualOne -- Parts equal to 1.
PartsEqualToMultiplicity -- Parts equal to their own multiplicity.
PartsEqualToNextPart -- Parts whose next smaller part has the same size: the number of parts less the distinct parts.
PartsEqualTwo -- Parts equal to 2.
PartsExceedingOnes -- Parts larger than the number of parts equal to 1.
PartsNotCongruentToOneModuloThree -- Parts that are not congruent to 1 modulo 3.
PartsNotCongruentToTwoModuloThree -- Parts that are not congruent to 2 modulo 3.
PartsNotCongruentToZeroModuloThree -- Parts that are not divisible by 3.
PlancherelCount -- The square of the number of standard tableaux: the permutations whose Robinson-Schensted shape is this partition.
PowerSumSymmetricFunctionAtOne -- The power-sum symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the number of parts to its own power.
RefinementCount -- The partitions that refine this one, whose parts subdivide to give theirs. Itself included.
RemovableHorizontalDominoes -- Parts from which 2 can be subtracted and leave a partition: rows ending in a removable horizontal domino.
RimHooksRemovedToThreeCore -- The 3-rim hooks removed to reach the 3-core.
RimHooksRemovedToTwoCore -- The 2-rim hooks removed to reach the 2-core: the size less the core's, over 2, the core read off the abacus.
RookPlacementCount -- The rook placements, of any number of rooks, on the Ferrers board of the diagram.
RowCellPairs -- Pairs of cells in the same row: the sum over parts of the binomial coefficient of the part and 2.
RowRemovalGrundyValue -- The Grundy value of the game that removes cells from the end of a row while the diagram stays a partition: staircase nim, the nim-sum of the differences of consecutive parts at odd positions.
SameLengthSubpartitionCount -- The partitions with as many parts contained in this one.
SeaweedAlgebraDimension -- The dimension of the maximal parabolic seaweed algebra: half the sum of the squares of the parts and of the size.
SecondAlladiWeight -- Half the last part (rounded up if odd), times the product over consecutive parts of half the gap less 1 for each even one of the two.
SemistandardTableauCountWithEntriesUpToFour -- Semistandard tableaux of the shape with entries at most 4.
SemistandardTableauCountWithEntriesUpToLength -- Semistandard tableaux of the shape with entries at most the number of parts.
SemistandardTableauCountWithEntriesUpToSize -- Semistandard tableaux of the shape with entries at most the size of the partition.
SemistandardTableauCountWithEntriesUpToThree -- Semistandard tableaux of the shape with entries at most 3.
SemistandardTableauCountWithEntriesUpToTwo -- Semistandard tableaux of the shape with entries at most 2: the product of 2 + j - i over the cells, over the hook product.
SetPartitionsWithBlockSizesCount -- The number of set partitions whose block sizes are the parts: the size factorial over the product of the factorials of the parts and of the multiplicities.
ShiftedStandardTableauCount -- Standard tableaux of the shifted shape: the multinomial of the parts times, over pairs of parts, the difference over the sum. 0 when two parts are equal.
ShorterSideRookPlacementCount -- The ways to place as many non-attacking rooks as the shorter side of the Ferrers board's bounding rectangle has cells, 0 when they do not fit.
SingletonParts -- Parts that occur exactly once.
SingletonRefinementCount -- The chains of single splits (a part into two) from this partition down to all ones.
SmallestMissingPart -- The smallest positive integer that is not a part.
SmallestNonRepeatedInteger -- The smallest positive integer that is not a part at least twice.
SmirnovWordCount -- The words with each letter used this many times and no letter twice in a row: the Carlitz-Scoville-Vaughan sum over the numbers of runs of each letter, taken as a binomial convolution over the parts.
StandardTableauCount -- The number of standard Young tableaux of the shape: the size factorial over the hook product.
StandardTableauCountWithMajorIndexCongruentToOneModuloSize -- The standard Young tableaux whose major index leaves remainder 1 on division by the size of the shape (never, when the size is 1).
StarTranspositionFactorizationCount -- Minimal transitive factorizations of a permutation of this cycle type into star transpositions: the product of the parts times (n + 1) ... (n + length - 2).
SubpartitionCount -- The partitions contained in this one.
SubsetGreatestCommonDivisorSum -- The sum, over the non-empty subsets of the parts, of their greatest common divisor: the sum over d of phi(d) times one less than 2 to the number of parts divisible by d.
SumOfEvenParts -- The sum of the even parts.
SumOfHookLengths -- The total of all hook lengths.
SumOfOddParts -- The sum of the odd parts.
SumOfPartsAtLeastTwo -- The sum of the parts that are at least 2.
SymmetricDiagonalHooks -- Cells on the main diagonal whose arm and leg are equal.
SymplecticDimensionRankThree -- The dimension of the irreducible representation of Sp(6) with this highest weight; 0 if the partition has more than 3 parts.
SymplecticDimensionRankTwo -- The dimension of the irreducible representation of Sp(4) with this highest weight; 0 if the partition has more than 2 parts.
TransitiveMonotoneFactorizationCount -- Transitive monotone factorizations of genus zero of a permutation of this cycle type, into n + length - 2 transpositions.
TwelveTimesMajorIndexVarianceOfStandardTableaux -- Twelve times the variance of the major index over the standard tableaux: the sum of the first n squares, less the sum of the squared hook lengths.
TwiceMeanMajorIndexOfStandardTableaux -- Twice the mean major index over the standard tableaux: n choose 2, less the sum of the row lengths choose 2, plus the sum of the column lengths choose 2.
UpperCoverCountInDominanceOrder -- The partitions of the same size that cover this one in dominance order.
YoungLatticeCoveringRelationCount -- The covering relations of Young's lattice among the partitions contained in this one: the sum of their corners.

Path algebras

LinearQuiver -- The quiver $A_n$ : $1 \to 2 \to \cdots \to n$ , acyclic, whose path algebra is the incidence algebra of a chain.
PathAlgebra -- The path algebra $kQ$ of a quiver: basis QuiverPath, product concatenation, finite-dimensional exactly when the quiver is acyclic.
Quiver -- A directed multigraph named by its vertex count and its arrows as $(\text{from},\text{to})$ pairs — loops and parallel arrows allowed.
QuiverCompose -- Concatenate two paths — or zero, when $q$ does not start where $p$ ends. Most products in a path algebra are zero, and that is the structure rather than a failure.
QuiverIsAcyclic -- Whether the quiver has no directed cycle — equivalently, whether its path algebra is finite-dimensional at all.
QuiverPath -- A directed path in a quiver: where it starts, and the arrow indices it follows. The basis of the path algebra $kQ$ , with one trivial path per vertex.
QuiverPathEnd -- The vertex a path ends at, following its arrows through the quiver from its start.

Path and tree statistics

Height -- The height of a Dyck path, a binary tree or an ordered tree, each counted as FindStat defines it.

Perfect matching statistics

AlignedPairCount -- Pairs of pairs (a, b), (c, d) with a < b < c < d.
ClosersBelowLargestOpener -- The pairs whose larger point is below the largest opener, the larger point of the pair opening last.
CrossingsPlusTwiceNestings -- The crossings plus twice the nestings.
EarlyCloserPairCount -- Pairs whose larger point is at most n + 1, for a matching of 2n points.
IdentityMatchingEdgeCount -- Pairs {2k - 1, 2k}: the edges shared with the matching 12, 34, 56, ....
MaximalNestingCount -- The most pairs that nest one pair: the largest number of pairs (a, b) with a < c and d < b, over the pairs (c, d).
MaximalRightCrossingCount -- The most pairs (c, d) with a < c < b < d, over the pairs (a, b): arcs that cross a given arc at its right end.
PropagatingNumber -- Pairs with one point among the first n and the other among the last n, for a matching of 2n points.

Permutation statistics 1 ≝1 primitive

Adjacencies -- Positions i with p(i+1) = p(i) - 1.
AdjacenciesWithZeroAppended -- Positions i with p(i) = p(i+1) + 1, reading p(n+1) = 0.
AdjacentCycles -- Cycles of the form (a, a+1, ..., a+k), fixed points included.
AdjacentTranspositions -- Two-cycles (i, i+1).
AdmissibleInversions -- Pairs i < j with p(i) > p(j) and either p(i-1) < p(i) or some entry between them larger than p(i).
AdmissibleInversionsPlusDescents -- ShareshianWachsAdmissibleInversions plus the descents.
Alignments -- Pairs i, j of arcs of the permutation diagram that align.
AlternatingRuns -- The interior peaks plus the interior valleys.
Antiexcedances -- The number of antiexcedances of $p$ : positions $i$ with $p_i < i$ .
ArithmeticProgressionsOfLengthTwo -- Triples i < j < k with p(k) - p(j) = p(j) - p(i).
AscentBottomSum -- The sum of the ascent bottoms: p(i) over the ascents i.
AscentTopSum -- The sum of the ascent tops: p(i+1) over the ascents i.
AscentTopsWithSmallerValuesBefore -- Ascents i whose top p(i+1) has every smaller value among p(1), ..., p(i).
Ascents -- The number of ascents of $p$ : positions $i$ with $p_i < p_{i+1}$ .
AscentsOfDistanceTwo -- Positions i with p(i) < p(i+2).
AugmentedDoubleAscents -- Positions i with w(i) < w(i+1) < w(i+2) in the word w = 0 p(1) ... p(n).
BabsonSteingrimssonStatistic -- Babson and Steingrimsson's stat: (13-2) + (21-3) + (32-1) + (21).
BabsonSteingrimssonStatisticDoublePrime -- Descents plus the occurrences of the vincular patterns 1-32, 3-12 and 3-21.
BabsonSteingrimssonStatisticPrime -- Descents plus the occurrences of the vincular patterns 13-2, 31-2 and 32-1.
BigAscents -- Positions i with p(i+1) - p(i) > 1.
BigAscentsAfterPrependingZero -- Positions of the word 0 p(1) ... p(n) that rise by more than 1, the first entry counted against 0.
BigDeficiencies -- Positions with i - p(i) > 1.
BigDescents -- Positions i with p(i) - p(i+1) > 1.
BigExcedances -- Positions with p(i) - i > 1.
BinStatistic -- The sum of the binomial coefficients (i + 1 choose 2) over the descents i.
Bonds -- Positions i with |p(i) - p(i+1)| = 1.
BoxedOccurrencesOf123 -- Occurrences of 123 with no other entry positioned between the first and last and valued between them.
BoxedOccurrencesOf132 -- Occurrences of 132 with no other entry positioned between the first and last and valued between the least and greatest.
BruhatLowerCovers -- Pairs i < j with p(i) > p(j) and no entry between them valued between p(j) and p(i).
ComajorIndex -- The sum of n - i over the descents i.
ConjugacyClassSize -- The number of permutations of the same cycle type: n! over the product of the cycle sizes and the factorials of the multiplicities.
Convexity -- The largest 2 p(i) - p(i-1) - p(i+1) over the interior positions, or 0.
CycleCountkernel -- The number of cycles in the disjoint-cycle decomposition of $p$ .
CycleDescents -- Steps down within the cycles, each written from its smallest point.
CycleMinimumExceedancesDeficiencies -- The sum over the cycles of the lesser of the number of exceedances and the number of the other points.
CyclePeaks -- Values i with p^-1(i) < i > p(i).
CyclesOfLengthAtLeastThree -- Cycles of size at least three.
CyclesOfLengthAtMostThree -- Cycles of size at most three.
CyclesOfLengthAtMostTwo -- Cycles of size one or two.
CyclicAlignments -- Pairs (i, j) with i, j, p(j), p(i) in cyclic order.
CyclicBonds -- Positions i with |p(i) - p(i+1)| = 1, with p(n+1) = p(1).
CyclicCrossings -- Pairs (i, j) with i, p(j), p(i), j in cyclic order.
CyclicDescents -- Descents of p read cyclically, counting position n when p(n) > p(1).
CyclicMaximalJump -- The largest |p(i+1) - p(i)|, with p(n+1) = p(1).
CyclicMisalignments -- Pairs (i, j) with i, p(i), j, p(j) in cyclic order.
CyclicSmallExcedances -- Positions with p(i) = i + 1, read cyclically: the last position counts when p(n) = 1.
CyclicSmallWeakExcedances -- Positions with p(i) = i or p(i) = i + 1, read cyclically.
CyclicValleys -- Positions lower than both neighbours, reading the word cyclically.
CyclicallyAdjacentTranspositions -- Two-cycles (i, i+1), and the two-cycle (1, n).
DecompositionNumber -- The k with {p(1), ..., p(k)} = {1, ..., k}: the blocks of p as a direct sum.
Denert -- Sum of the excedance positions, plus the inversions within each of the excedance and non-excedance subwords.
Depth -- Half the total displacement, (1/2) * sum |p(i) - i|.
DescentBottomSum -- The sum of the descent bottoms: p(i+1) over the descents i.
DescentDifferenceSum -- The sum of p(i) - p(i+1) over the descents i.
DescentTopSum -- The sum of the descent tops: p(i) over the descents i.
DescentWeightMinusInversions -- The sum of i(n - i) over the descents i, less the inversions.
Descents -- The number of descents of $p$ : positions $i$ with $p_i > p_{i+1}$ .
DescentsAfterZeroingFixedPoints -- The descents of the word with every fixed point replaced by 0.
DescentsAndRecoils -- Indices i < n that are both a descent of p and a recoil.
DescentsMinusOneUnlessFirstEntryOne -- The number of descents of p, less one when p(1) is not 1.
DescentsOfDistanceThree -- Positions i with p(i) > p(i+3).
DescentsOfDistanceTwo -- Positions i with p(i) > p(i+2).
DescentsOfTypeTwo -- Descents i with p(i) larger than everything before it.
DescentsOrRecoils -- Indices i < n that are a descent of p or a recoil.
DescentsPlusRecoils -- Descents plus recoils: the two counts added, not the union of the indices.
DiaconisGrahamDefect -- Half of the total displacement less the number of inversions and the reflection length.
Disorder -- Scanning cyclically for 1, then 2, ...: the entries passed that are still larger than the one sought.
DistinctCycleLengths -- How many distinct cycle sizes occur.
DistinctDiagonalSums -- The number of distinct sums along the diagonals of the permutation matrix.
DoubleAscents -- Positions i with p(i) < p(i+1) < p(i+2).
DoubleDeficiencies -- Positions with i > p(i) > p(p(i)).
DoubleDescents -- Positions i with p(i) > p(i+1) > p(i+2).
DoubleExcedances -- Positions with i < p(i) < p(p(i)).
DoubleSuccessions -- Positions i with p(i+1) = p(i) + 1 and p(i+2) = p(i) + 2.
ElizaldePakRank -- The largest k with p(i) > k for all i <= k.
EvenDescents -- Descents i with i even.
EvenInversions -- Inversions i < j with j - i even.
EvenWeakDeficienciesAndOddWeakExcedances -- Positions i with p(i) <= i for even i, and p(i) >= i for odd i.
Excedances -- The number of excedances of $p$ : positions $i$ with $p_i > i$ .
ExclusiveLeftToRightMaxima -- Left-to-right maxima that are not right-to-left minima.
ExclusiveRightToLeftMinima -- Right-to-left minima that are not left-to-right maxima.
ExteriorLength -- The longest proper prefix whose relative order is that of the suffix of the same length.
FinalRises -- The ascents at the end: n - 1 less the last descent position.
FirstAscent -- The smallest ascent position, or n when p is decreasing.
FirstDescent -- The smallest descent position, or n when p is increasing.
FirstDirectSummandSize -- The smallest k with {p(1), ..., p(k)} = {1, ..., k}.
FirstEntry -- The first entry p(1).
FixedPointMajorIndex -- The sum of the fixed points less the triangular number of their count, plus the major index of the word without its fixed points.
FixedPoints -- The number of fixed points of $p$ : positions $i$ with $p_i = i$ .
FixedPointsBeforeLargestMovedPoint -- Fixed points smaller than the largest i with p(i) != i.
GlobalAscents -- Positions i < n with p(j) < p(k) whenever j <= i < k.
GlobalDescents -- Positions i < n with p(j) > p(k) whenever j <= i < k.
GreaterNeighborOfMaximum -- The larger of the two entries beside n, an end counting as 0.
Hag -- The excedance tops less their positions, plus the inversions among the excedance tops less those among the rest, plus the pairs k < i, over excedances i, with p(k) <= i.
HalfParityMismatches -- Half the positions i where i and p(i) differ in parity.
HalfPermanentOfIdentityPlusPermutation -- Half the permanent of the identity matrix plus the permutation matrix, which is 2 to the power of the cycles less one.
HertzsprungOccurrencesOf132 -- Positions i with p(i+1) = p(i) + 2 and p(i+2) = p(i) + 1.
HookFactorInversions -- The inversions summed over the hook factors t1, ..., tk of p.
HookFactorizationPrefixLength -- The length of the leading increasing factor p of the hook factorization p t1 ... tk.
InterpolatingPolynomialDegree -- The degree of the polynomial through the points (i, p(i)): the largest k with a nonzero k-th finite difference.
InverseMajorIndex -- The major index of the inverse: the sum of the values v with v + 1 to the left of v.
InversionIndex -- The sum of the larger entry p(i) over the inversions i < j.
InversionSum -- The sum of j - i over the inversions i < j.
Inversions -- The number of inversions of a permutation $p$ : pairs $i < j$ with $p_i > p_j$ (its Kendall-tau distance from the identity).
InversionsBetweenExcedancesAndFixedPoints -- Pairs i < j = p(j) < p(i).
InversionsBetweenExcedancesWithLinkedGreater -- Pairs i < j < p(j) < p(i) with p^-1(j) < j.
InversionsOfCyclicEmbedding -- The inversions of the cycle (p(1), ..., p(n), n + 1), as a permutation of n + 1.
InversionsOfDistanceAtMostThree -- Inversions i < j with j - i <= 3.
InversionsOfDistanceAtMostTwo -- Inversions i < j with j - i <= 2.
InversionsOfSecondEntry -- Positions after the second with a smaller entry than p(2).
InversionsOfThirdEntry -- Positions after the third with a smaller entry than p(3).
InversionsPlusMajorIndex -- The number of inversions plus the major index.
InvisibleDescents -- Positions i with p(i) > p(i+1) > i.
InvisibleInversions -- Pairs i < j with p(i) > p(j) > i.
IsolatedDescents -- Descents i such that neither i-1 nor i+1 is a descent.
LargestCycleLength -- The size of the largest cycle.
LargestDirectSummandSize -- The size of the largest block of p as a direct sum.
LargestGreatestCommonDivisorOfPositionAndValue -- The largest gcd(i, p(i)).
LargestRunLength -- The length of the longest increasing run (the catalog's second spelling of LongestRun).
LargestSearchTreeLeafLabel -- The largest label of a leaf of the binary search tree of p, inserting p(1), ..., p(n) in order.
LargestWeakExcedancePosition -- The largest i with p(i) >= i.
LastDescent -- The largest descent position, or 0 when p is increasing.
LastEntry -- The last entry p(n).
LeftOuterPeaks -- Peaks of the word 0 p(1) ... p(n): the interior peaks, plus one when p(1) > p(2).
LeftToRightMaxima -- Positions larger than everything before them.
LeftToRightMaximaOrRightToLeftMinima -- Positions that are a left-to-right maximum, a right-to-left minimum, or both.
LeftToRightMaximaPositionSum -- The sum of the positions of the left-to-right maxima.
LeftToRightMaximaXorRightToLeftMinima -- Positions that are a left-to-right maximum or a right-to-left minimum but not both.
LeftToRightMinima -- Positions smaller than everything before them.
LexicographicRank -- The position of p among the permutations of its size in lexicographic order, counting from 1.
Load -- The major index of the reverse of the inverse.
LongestArithmeticProgression -- The length of a longest subsequence whose entries form an arithmetic progression, of either sign.
LongestCycleLength -- The size of the largest cycle (the catalog's second spelling).
LongestDecreasingSubsequence -- The length of a longest decreasing subsequence.
LongestIncreasingSubsequence -- The length of a longest increasing subsequence.
LongestIncreasingSubsequenceCount -- The number of longest increasing subsequences of p.
LongestKOneTwoPattern -- The largest k such that p has an occurrence of k 1 2 ... (k-1): a first entry above an increasing run of later, smaller ones.
LongestRun -- The length of the longest run of consecutive increases.
LongestSuccessionRun -- The length of the longest factor p(k), ..., p(l) with p(i+1) = p(i) + 1.
LowerMiddleEntry -- The entry p(floor((n+1)/2)).
Mad -- Foata and Zeilberger's mad: 2 (2-31) + (31-2) + (21).
MajorIndex≝ -- The major index of $p$ : the sum of the descent positions $\sum_{p_i > p_{i+1}} i$ .
MajorIndexAfterZeroingFixedPoints -- The major index of the word with every fixed point replaced by 0.
MajorIndexMinusExcedances -- The major index less the number of positions with p(i) > i.
MajorIndexMinusInversions -- The major index less the number of inversions.
MajorIndexPlusInverseMajorIndex -- The major index plus the major index of the inverse.
Mak -- Foata and Zeilberger's mak: (2-31) + (32-1) + (1-32) + (21).
Makl -- Foata and Zeilberger's makl: (1-32) + (31-2) + (32-1) + (21).
MaximalCycleDifference -- The largest difference between two elements of one cycle.
MaximalDiagonalOccupancy -- The most entries on one diagonal of the permutation matrix: the largest number of i sharing a p(i) - i.
MaximalDisplacement -- The largest |p(i) - i|.
MaximalExcedance -- The largest p(i) - i.
MaximalJump -- The largest |p(i+1) - p(i)|.
MaximalModularDisplacement -- The largest cyclic distance between p(i) and i, taken mod n.
MaximalRiseSize -- The largest p(i+1) - p(i), or 0 when p has no rise.
MaximalSuccessiveDifference -- The largest p(i+1) - p(i), negative for a decreasing permutation.
MaximumOfDescentsAndInverseDescents -- The larger of the number of descents and the number of inverse descents.
MidPointsOfOccurrencesOf123 -- Positions that are the middle of a 123 occurrence.
MidPointsOfOccurrencesOf321 -- Positions that are the middle of a 321 occurrence: neither a left-to-right maximum nor a right-to-left minimum.
MinimalCycleDifference -- The smallest, over the cycles, of the largest difference between two elements of a cycle.
MinimalInversionsEitherDirection -- The smaller of the number of inversions and the number of non-inversions.
MinimalJump -- The smallest |p(i+1) - p(i)|.
MinorIndex -- The minor index of $p$ : the sum of the ascent positions $\sum_{p_i < p_{i+1}} i$ .
NeighboringNumber -- The smallest |p(i+1) - p(i)|, the wrap-around gap |p(1) - p(n)| included.
NestedExcedances -- Pairs i < j with i < j < p(j) < p(i).
NonAttackingNeighbors -- Positions i with |p(i) - p(i+1)| > 1.
NonCyclicSmallExcedances -- Positions that are not cyclic small excedances.
NonCyclicSmallWeakExcedances -- Positions that are not cyclic small weak excedances.
NonFixedPoints -- Positions i with p(i) != i.
NonInitialLeftToRightMinima -- Positions i >= 2 with p(i) smaller than everything before it.
NonInversionSum -- The sum of j - i over the non-inversions i < j.
NonInversions -- Pairs i < j with p(i) < p(j).
NonLeftToRightMaxima -- Positions smaller than something before them.
NonNegativeWalkSteps -- Steps of the walk (up at an ascent, down at a descent) that begin and end at height at least 0.
NonOverlappingDescents -- Descents taken greedily left to right, none starting where the last one ended.
NonRecords -- Positions that are none of a left-to-right maximum or minimum, or a right-to-left maximum or minimum.
NonSmallExcedances -- Positions with p(i) != i + 1.
NonSmallWeakExcedances -- Positions with p(i) not in {i, i + 1}.
NonWeakExcedanceMidPointsOfOccurrencesOf321 -- Positions i with p(i) < i that are the middle of a 321 occurrence.
NontrivialCycles -- Cycles of size at least two: the cycles less the fixed points.
NormalizedLeafLabelSum -- The sum of the leaf labels of the increasing binary tree of p, less n.
NormalizedSumOfDistancesToGreaterElement -- The same sum, each gap one less: the entries strictly between a position and its nearest greater entry.
OccurrencesOf123 -- Triples i < j < k with p(i) < p(j) < p(k).
OccurrencesOf123Or132 -- Triples ordered as 123 or as 132.
OccurrencesOf123Or213 -- Triples ordered as 123 or as 213.
OccurrencesOf123Or231 -- Triples ordered as 123 or as 231.
OccurrencesOf123Or312 -- Triples ordered as 123 or as 312.
OccurrencesOf123Or321 -- Triples ordered as 123 or as 321.
OccurrencesOf12Dash3 -- Triples i, i+1, k with k > i+1 ordered as 123 (12-3).
OccurrencesOf132 -- Triples i < j < k with p(i) < p(k) < p(j).
OccurrencesOf1324 -- Quadruples i < j < k < l ordered as 1324.
OccurrencesOf132Or213 -- Triples ordered as 132 or as 213.
OccurrencesOf132Or213Or321 -- Triples ordered as 132, 213 or 321.
OccurrencesOf132Or231 -- Triples ordered as 132 or as 231.
OccurrencesOf132Or312 -- Triples ordered as 132 or as 312.
OccurrencesOf132Or321 -- Triples ordered as 132 or as 321.
OccurrencesOf13Dash2 -- Triples i, i+1, k with k > i+1 ordered as 132 (13-2).
OccurrencesOf1432 -- Quadruples i < j < k < l ordered as 1432.
OccurrencesOf14Dash2Dash3Or14Dash3Dash2 -- Adjacent ascents p(i) < p(i+1) with two later entries between them, in either order, counted in pairs.
OccurrencesOf213 -- Triples i < j < k with p(j) < p(i) < p(k).
OccurrencesOf213Or231 -- Triples ordered as 213 or as 231.
OccurrencesOf213Or312 -- Triples ordered as 213 or as 312.
OccurrencesOf213Or321 -- Triples ordered as 213 or as 321.
OccurrencesOf2143 -- Quadruples i < j < k < l ordered as 2143.
OccurrencesOf2143Or4213 -- Quadruples ordered as 2143 or as 4213.
OccurrencesOf21Dash3 -- Triples i, i+1, k with k > i+1 ordered as 213 (21-3).
OccurrencesOf231 -- Triples i < j < k with p(k) < p(i) < p(j).
OccurrencesOf231Or312 -- Triples ordered as 231 or as 312.
OccurrencesOf231Or321 -- Triples ordered as 231 or as 321.
OccurrencesOf23Dash1 -- Triples i, i+1, k with k > i+1 ordered as 231 (23-1).
OccurrencesOf2413 -- Quadruples i < j < k < l ordered as 2413.
OccurrencesOf312 -- Triples i < j < k with p(j) < p(k) < p(i).
OccurrencesOf312Or321 -- Triples ordered as 312 or as 321.
OccurrencesOf31Dash2 -- Triples i, i+1, k with k > i+1 ordered as 312 (31-2).
OccurrencesOf321 -- Triples i < j < k with p(i) > p(j) > p(k).
OccurrencesOf321Or3412 -- Triples ordered as 321, plus quadruples ordered as 3412.
OccurrencesOf3241 -- Quadruples i < j < k < l ordered as 3241.
OccurrencesOf3241Or4231 -- Quadruples ordered as 3241 or as 4231.
OccurrencesOf32Dash1 -- Triples i, i+1, k with k > i+1 ordered as 321 (32-1).
OccurrencesOf3412Or4231 -- Quadruples ordered as 3412 or as 4231.
OccurrencesOf4132Or4231 -- Quadruples ordered as 4132 or as 4231.
OccurrencesOf41352Or42351Or51342Or52341 -- Quintuples ordered as 41352, 42351, 51342 or 52341.
OccurrencesOf4213 -- Quadruples i < j < k < l ordered as 4213.
OccurrencesOf4231 -- Quadruples i < j < k < l ordered as 4231.
OccurrencesOf52341 -- Quintuples i < j < k < l < m ordered as 52341.
OccurrencesOfConsecutive132 -- Triples i, i+1, i+2 ordered as 132 (consecutive 132).
OccurrencesOfInitial123 -- Triples 1 < j < k ordered as 123 (|123).
OccurrencesOfInitial12Dash3 -- Triples 1, 2, k ordered as 123 (|12-3).
OccurrencesOfInitial132 -- Triples 1 < j < k ordered as 132 (|132).
OccurrencesOfInitial1Dash23 -- Triples 1, j, j+1 ordered as 123 (|1-23).
OccurrencesOfInitial213 -- Triples 1 < j < k ordered as 213 (|213).
OccurrencesOfInitial231 -- Triples 1 < j < k ordered as 231 (|231).
OccurrencesOfInitial2Dash13 -- Triples 1, j, j+1 ordered as 213 (|2-13).
OccurrencesOfInitial312 -- Triples 1 < j < k ordered as 312 (|312).
OccurrencesOfInitial321 -- Triples 1 < j < k ordered as 321 (|321).
OddDescents -- Descents i with i odd.
OddInversions -- Inversions i < j with j - i odd.
OneBoxPatterns -- Positions i with a neighbouring entry, left or right, differing from p(i) by exactly 1.
OuterPeaks -- Peaks of the word 0 p(1) ... p(n) 0: the interior peaks, plus one for each end that is a peak.
OverlapSetSize -- Shifts i < n for which the first n - i entries and the last n - i entries are in the same relative order.
Peaks -- The number of peaks of $p$ : interior positions $i$ with $p_{i-1} < p_i > p_{i+1}$ .
PermutationBounce -- The sum of n - b over the bounce points b: the position of 1 in p, then the greatest position among the entries up to one past the last.
PermutationBreadth -- The least Manhattan distance |i - j| + |p(i) - p(j)| between two points of the permutation matrix.
PermutationCosine -- The sum of i p(i).
PermutationGenus -- The genus g with n + 1 - 2g the cycles of p plus the cycles of p^-1 composed with the long cycle.
PermutationOrder -- The order of p in the symmetric group — the lcm of its cycle lengths.
PermutationSupport -- The points $p$ moves, in increasing order.
PermutationTreeHeight -- The height of the rooted tree from inserting the entries one by one, the most entries on the stack of ancestors.
PinnacleSum -- The sum of the peak values: p(i) over the interior peaks i.
PositionsOfLetterOneIn123 -- Positions that start an occurrence of 123.
PositionsOfLetterOneIn132 -- Positions that start an occurrence of 132.
PositionsOfLetterThreeIn132 -- Positions that are the middle of an occurrence of 132.
PositionsOfLetterTwoIn213 -- Positions that start an occurrence of 213.
PrefixSumProduct -- The product of the prefix sums p(1) + ... + p(i) for i < n.
PrimeColumnSums -- Positions i with i + p(i) prime.
PrimeProductsMinusOne -- Positions i with i * p(i) - 1 prime.
PrimeProductsPlusOne -- Positions i with i * p(i) + 1 prime.
ProperBlocks -- Windows p(i), ..., p(j) of two or more entries whose values are consecutive integers, the whole word excepted.
PureExcedances -- Excedances i < p(i) with no j < i where i <= p(j) < p(i).
Rafts -- Maximal runs of consecutive successions p(i+1) = p(i) + 1.
Recoils -- Values i with i + 1 to the left of i: the descents of the inverse.
Records -- The number of records (left-to-right maxima) of $p$ : positions $i$ with $p_i > p_j$ for all $j < i$ .
RecursiveEndMaximumCount -- FindStat's rix: the number of times, peeling the word, the maximum of what is left is its last entry.
RecursiveStartMinimumCount -- FindStat's aix: the number of times, peeling the word, the minimum of what is left is its first entry.
ReducedWordSupport -- The simple reflections in a reduced word: the i < n with some entry of p(1), ..., p(i) larger than i.
ReflectionLength -- n minus the number of cycles — the minimum number of transpositions.
RepeatedLehmerCodeEntries -- The Lehmer code's length less its number of distinct entries.
RestrictedNonInversionsBetweenExcedances -- Pairs i < j < p(i) < p(j).
RestrictedNonInversionsBetweenExcedancesWithLinkedRightmost -- Pairs i < j < p(i) < p(j) with p^-1(j) < j.
RightFloats -- Large ascents p(i+1) > p(i) + 1 beside no raft.
RightOuterPeaks -- Peaks of the word p(1) ... p(n) 0: the interior peaks, plus one when p(n) > p(n-1).
RightRopes -- Large ascents right after a raft.
RightTethers -- Large ascents between two rafts.
RightToLeftMaxima -- Positions larger than everything after them.
RightToLeftMinima -- Positions smaller than everything after them.
Runs -- Maximal increasing runs — one more than the number of descents.
ScaledReciprocalPositionValueSum -- The sum of 1/(i + p(i)) times the least common multiple of 1 .. 2n, an integer.
Separators -- The values v that are a vertical separator of p, or whose image under p^-1 is a vertical separator of p^-1.
ShareshianWachsAdmissibleInversions -- Pairs i < j with p(i) > p(j) and either p(j) < p(j+1) or some entry between them smaller than p(j).
ShiftedMajorIndex -- The sum of the positions i with p(i) - p(i+1) >= 2.
ShortestCycleLength -- The size of the smallest cycle.
SimsunDoubleDescents -- Double descents in the restrictions of p to the values 1..k, over all k, counted with multiplicity.
SmallExcedances -- Positions with p(i) = i + 1.
SmallWeakExcedances -- Positions with p(i) = i or p(i) = i + 1.
SmallerChildPathLeafLabel -- The label of the leaf reached from the root of the increasing binary tree of p by always following the child with the smaller label.
SmallestFixedPoint -- The smallest i with p(i) = i, or n + 1 when there is none.
SmallestLeafLabel -- The smallest label of a leaf of the increasing binary tree of p: the least peak, an end counting as a peak.
SortingIndex -- The total distance the entries move in selection sort.
StackSortable -- 1 when p avoids the pattern 231, 0 otherwise — exactly the stack-sortable permutations.
StaircaseSizeOfCode -- The largest k with a subsequence of the Lehmer code whose j-th entry from the right is at least j.
StandardizedBiAlternatingInversionNumber -- Half of floor(n/2)^2 plus the signed inversions, each pair i < j weighted by (-1)^(i+j).
StretchingPairs -- Pairs i < j with p(i) < i < j < p(j).
StrongFixedPoints -- Positions that are both a left-to-right maximum and a right-to-left minimum.
Successions -- Permutation: positions i with p(i+1) = p(i) + 1. SetPartition: elements i with i + 1 in the same block.
SumOfDistancesToGreaterElement -- Over each position, the gap to the nearest greater entry, the ends counting as greater.
SymmetryClassSize -- The permutations reachable from p by inverse, reverse and complement: 8 over the symmetries that fix p.
TailPairsOfOccurrencesOf321 -- Pairs j < k with p(j) > p(k) and some entry before j above p(j), the positions of the 2 and the 1 of the occurrences of 321.
ThreeCycleCount -- Cycles of size exactly three.
ThreeExcedances -- Positions with p(i) = i + 3.
ThreeRises -- Positions i with p(i+1) = p(i) + 3.
ThreeShiftedMajorIndex -- The sum of the positions i with p(i) - p(i+1) > 3.
Tier -- Positions a with a larger entry, then the value p(a) - 1, to its right.
TotalDisplacement -- The sum of |p(i) - i|.
TotalSquaredDisplacement -- The sum of (p(i) - i)^2.
TwoCycleCount -- Cycles of size exactly two.
TwoExcedances -- Positions with p(i) = i + 2.
TwoRises -- Positions i with p(i+1) = p(i) + 2.
TwoShiftedMajorIndex -- The sum of the positions i with p(i) - p(i+1) > 2.
UlamDistance -- The entries to remove to leave an increasing sequence: n less the length of a longest increasing subsequence.
UpDownRuns -- Maximal monotone runs, with p(1) counted alone when it is a descent.
UpperMiddleEntry -- The entry p(floor(n/2) + 1).
Valleys -- The number of valleys of $p$ : interior positions $i$ with $p_{i-1} > p_i < p_{i+1}$ .
ValleysIncludingBoundary -- Positions lower than both neighbours, a larger entry standing off each end.
VerticalSeparators -- Interior positions i with |p(i-1) - p(i+1)| = 1.
VeryBigAscents -- Positions i with p(i+1) - p(i) > 2.
VeryBigDescents -- Positions i with p(i) - p(i+1) > 2.
VisibleDescents -- Positions i with p(i+1) <= min(i, p(i)).
VisibleInversions -- Pairs i < j with p(j) <= min(i, p(i)).
WeakAntiexcedances -- Positions with p(i) <= i.
WeakExcedanceMidPointsOfOccurrencesOf321 -- Positions i with p(i) >= i that are the middle of a 321 occurrence.
WeakExcedances -- Positions with p(i) >= i.

Permutations

AlternatingGroup -- The alternating group $A_n$ : the even permutations of $\{1, \dots, n\}$ , of order $n!/2$ . A pure carrier like SymmetricGroup — GroupOrder, GroupElements and GroupGenerators read it directly, without materialising a Cayley table.
CycleIndexPolynomial -- The cycle index $Z(G) = \frac{1}{|G|}\sum_{g \in G} \prod_i x_i^{c_i(g)}$ of a permutation group, with $c_i(g)$ the number of $i$ -cycles of $g$ and $x_i$ the $i$ th variable. Evaluated at $x_i = k$ it counts the colorings with $k$ colors up to the group (Pólya).
Cycles -- A permutation written in disjoint-cycle notation: $(i_1\,i_2\,\dots\,i_k)$ sends $i_1 \to i_2 \to \dots \to i_k \to i_1$ and fixes everything else. A carrier, like PermutationGroup — it holds the cycles rather than computing anything from them.
GroupElements -- Every element of a group, or — for a PermutationGroup or AlternatingGroup with a second, list argument — the elements at those positions (Part-style, negative counts from the end).
GroupGenerators -- The generating permutations a PermutationGroup was built from, in Cycles notation — or, for AlternatingGroup, the fixed pair Wolfram itself uses.
GroupOrder -- $|G|$ : how many elements a group has. Works over every carrier this package knows — CyclicGroup, DihedralGroup, GroupDirectProduct, SymmetricGroup, AlternatingGroup and PermutationGroup.
InversePermutation -- The inverse $\sigma^{-1}$ of a permutation, in whichever notation it was given: a one-line word back to a one-line word, Cycles back to Cycles.
OrbitInventory -- Pólya's enumeration theorem. The inventory of the colorings a group acts on, up to the group, from a cycle index $Z$ and the weights of the colors, each $x_i$ becoming the power sum $p_i = \sum_j w_j^i$ . A single number $k$ for the weights gives the count of $k$ -colorings.
PermutationCycles -- A permutation given as a one-line word $\{\sigma(1), \dots, \sigma(n)\}$ , converted to disjoint-cycle notation (Cycles). The other direction of Permute's two conventions.
PermutationGroup -- The group generated by a list of permutations (Cycles), closed by breadth-first search over products of the generators. A carrier, like Cycles — GroupOrder, GroupElements and GroupGenerators read it directly rather than it computing anything itself.
PermutationList -- Cycles back to a one-line word: the image list $\{\sigma(1), \sigma(2), \dots\}$ . PermutationList(perm, n) pads that list with fixed points out to length $n$ .
PermutationPower -- The $n$ th power of a permutation under composition; a negative $n$ powers the inverse.
PermutationProduct -- The product of permutations, applying the left factor first. $(p\,q)(i) = q(p(i))$ .
PermutationReplace -- Apply a permutation by VALUE rather than by position: a point $i$ becomes $\sigma(i)$ , wherever it shows up. On a plain point or a list this renames entries in place; on Cycles it conjugates — $\sigma \cdot c \cdot \sigma^{-1}$ — since conjugating a cycle is exactly relabelling the points it names.
Permute -- Move the item at position $i$ of list to position $\sigma(i)$ , for a permutation perm given as Cycles or a one-line word. Given a PermutationGroup instead, returns list permuted by every element of the group.

Plane partition statistics

AmbientCubeSize -- The side of the smallest cube that holds the array: the most of its rows, its columns and its largest entry.
BoxCount -- The number of boxes: the sum of the entries.
ColumnCount -- The number of columns: the length of the first row.
ConjugateTrace -- The entries at least as large as their row number.
DurfeeCube -- The side of the largest cube in the array: the most diagonal entries d with the d-th at least d.
FirstRowSum -- The sum of the first row.
RemovableBoxCount -- The number of entries larger than both the entry to their right and the entry below.
RowCount -- The number of rows.

Relations

FindInstance -- n instances of vars that make expr true, over domain -- scoped to a provably correct core rather than Wolfram's full solver.
Inequality -- Wolfram's chained-comparison form -- v1 rel1 v2, v2 rel2 v3, ... all at once, letting the relation change partway through the chain (0 <= x < 5).

Sequences

BernoulliB -- The nth Bernoulli number, a rational sequence appearing in power-sum (Faulhaber) formulas and series expansions.
Fibonacci -- The nth Fibonacci number, with $F_0 = 0$ and $F_1 = 1$ , extended to negative n by the same recurrence.
LucasL -- The nth Lucas number: the Fibonacci-style recurrence started from $L_0 = 2$ , $L_1 = 1$ , closely related to Fibonacci.

Set partition statistics

Antisingletons -- Elements i with i + 1 in the same block, counting n + 1 as 1.
BlockMajorIndex -- The sum over blocks i of i times the elements of block i above the least element of block i + 1.
BlockSizeSpan -- Largest block size minus smallest.
Blocks -- The number of blocks.
BlocksAndAntisingletons -- Blocks plus antisingletons.
BlocksAtLeastTwo -- Blocks containing at least two elements.
BlocksSizeTwo -- Blocks containing exactly two elements.
BlocksWithEvenMinimum -- Blocks whose smallest element is even.
BlocksWithOddMinimum -- Blocks whose smallest element is odd.
CloserBeforeLargerPairs -- Pairs a < b in different blocks with a the greatest element of its block.
CloserBeforeOpenerPairs -- Pairs a < b in different blocks with a the greatest and b the least element of its block.
CloserBeforeSingletonPairs -- Pairs a < b in different blocks with a the greatest element of its block and b a singleton block.
CrossingNestingTotal -- Crossings plus nestings.
CrossingNumber -- The largest number of arcs that mutually cross.
Crossings -- SetPartition: pairs of arcs a < b < c < d with a~c and b~d. Permutation: pairs i, j with i < j <= p(i) < p(j) or p(i) < p(j) < i < j.
DepthIndex -- The sum of n - i over the arcs i, less the depth of each element, plus the depth of each arc.
DimensionExponent -- The sum over blocks of span minus (size - 1).
DimensionIndex -- The sum over blocks of largest minus smallest element, plus one.
DistinctBlockPairs -- Pairs of elements in different blocks.
FirstAtomicComponentBlockCount -- The least l for which the first l blocks hold exactly 1 .. m.
FirstBlockCardinality -- The size of the block containing 1.
FirstBlockSecondOrSmallestNonsingletonMaximum -- The smaller of the second element of the block holding 1 and the smallest closer; 1 when {1} is a block.
FlattenedAscentTopsAfterSmaller -- Ascent tops of the flattened word with every smaller element earlier.
FlattenedMajorIndex -- The major index of the permutation listing the blocks in order, each ascending.
HuangRhoadesMajorIndex -- The major index of the flattened permutation, plus for each block ending before the next begins the elements so far beyond each block's first.
InterlacingNumber -- Pairs of blocks whose i-th arcs interlace, each block closed by an arc to infinity.
InternalPoints -- Elements with an earlier and a later element whose blocks open after theirs does.
IntertwiningNumber -- Pairs a < b in different blocks with no element of either block between them.
LargestBlock -- The size of the largest block.
LargestElementInFirstBlock -- The largest element of the block containing 1.
LargestOpener -- The largest of the blocks' smallest elements.
LastBlockSize -- The size of the final block.
LeftCloserBigger -- Pairs of a closer j and a smaller element i of a later block.
LeftCloserSmaller -- Pairs of a closer j and a larger element i of a later block.
LeftOpenerSmaller -- Pairs of an opener j and a larger element i of a later block.
MahonianWordDescents -- Descents of the Mahonian word.
MahonianWordExcedances -- Excedances of the Mahonian word.
MahonianWordInversions -- Inversions of the Mahonian word.
MahonianWordMajorIndex -- The major index of the Mahonian word.
MahonianWordZIndex -- The major indices of the Mahonian word restricted to each pair of letters, summed.
MaximalArcLength -- The length of the longest arc; 0 when there is no arc.
MaximalBlockDifference -- The largest difference between two elements of one block.
MinimalArcLength -- The length of the shortest arc; the number of elements when there is no arc.
NestingNumber -- The largest number of arcs that mutually nest.
Nestings -- SetPartition: pairs of arcs a < b < c < d with a~d and b~c. Permutation: pairs i, j with j < i < p(i) < p(j) or p(j) < p(i) <= i < j.
OccurrencesOf12Over3 -- Occurrences of the pattern 12/3: a < b in one block and c > b in another.
OccurrencesOf12Over34 -- Occurrences of the pattern 12/34: a < b < c < d with a, b in one block and c, d in another.
OccurrencesOf13Over2 -- Occurrences of the pattern 13/2: a < c in one block with b between them in another.
OccurrencesOf13Over24 -- Occurrences of the pattern 13/24: a < b < c < d with a, c in one block and b, d in another.
OccurrencesOf13Over2Closers2 -- Occurrences of the pattern 13/2 with 2 the greatest element of its block.
OccurrencesOf13Over2Closers2Arc13 -- Occurrences of the pattern 13/2 with 2 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1 -- Occurrences of the pattern 13/2 with 1 the least element of its block.
OccurrencesOf13Over2Openers12Arc13 -- Occurrences of the pattern 13/2 with 1 and 2 the least elements of their blocks and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1Arc13 -- Occurrences of the pattern 13/2 with 1 the least element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1Closers2 -- Occurrences of the pattern 13/2 with 1 the least element of its block and 2 the greatest element of its block.
OccurrencesOf13Over2Openers1Closers3Arc13 -- Occurrences of the pattern 13/2 with 1 the least element of its block, 3 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers2Closers3Arc13 -- Occurrences of the pattern 13/2 with 2 the least element of its block, 3 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf1Over23 -- Occurrences of the pattern 1/23: b < c in one block and a < b in another.
OccurrencesOf1Over23Arc23 -- Occurrences of the pattern 1/23 with 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers1 -- Occurrences of the pattern 1/23 with 1 the greatest element of its block.
OccurrencesOf1Over23Closers13Arc23 -- Occurrences of the pattern 1/23 with 1 and 3 the greatest elements of their blocks and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers1Arc23 -- Occurrences of the pattern 1/23 with 1 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers3Arc23 -- Occurrences of the pattern 1/23 with 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1 -- Occurrences of the pattern 1/23 with 1 the least element of its block.
OccurrencesOf1Over23Openers12 -- Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks.
OccurrencesOf1Over23Openers12Arc23 -- Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers12Closers3Arc23 -- Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1Arc23 -- Occurrences of the pattern 1/23 with 1 the least element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1Closers3Arc23 -- Occurrences of the pattern 1/23 with 1 the least element of its block, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2 -- Occurrences of the pattern 1/23 with 2 the least element of its block.
OccurrencesOf1Over23Openers2Arc23 -- Occurrences of the pattern 1/23 with 2 the least element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2Closers1Arc23 -- Occurrences of the pattern 1/23 with 2 the least element of its block, 1 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2Closers3Arc23 -- Occurrences of the pattern 1/23 with 2 the least element of its block, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over2Over3 -- Occurrences of the pattern 1/2/3: a < b < c in three different blocks.
OccurrencesOf1Over2Over3Closers1 -- Occurrences of the pattern 1/2/3 with 1 the greatest element of its block.
OccurrencesOf1Over2Over3Closers13 -- Occurrences of the pattern 1/2/3 with 1 and 3 the greatest elements of their blocks.
OccurrencesOf1Over2Over3Closers2 -- Occurrences of the pattern 1/2/3 with 2 the greatest element of its block.
OccurrencesOf1Over2Over3Openers1 -- Occurrences of the pattern 1/2/3 with 1 the least element of its block.
OccurrencesOf1Over2Over3Openers12 -- Occurrences of the pattern 1/2/3 with 1 and 2 the least elements of their blocks.
OccurrencesOf1Over2Over3Openers12Closers3 -- Occurrences of the pattern 1/2/3 with 1 and 2 the least elements of their blocks and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers13Closers2 -- Occurrences of the pattern 1/2/3 with 1 and 3 the least elements of their blocks and 2 the greatest element of its block.
OccurrencesOf1Over2Over3Openers1Closers3 -- Occurrences of the pattern 1/2/3 with 1 the least element of its block and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers23 -- Occurrences of the pattern 1/2/3 with 2 and 3 the least elements of their blocks.
OccurrencesOf1Over2Over3Openers2Closers3 -- Occurrences of the pattern 1/2/3 with 2 the least element of its block and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers3Closers1 -- Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 the greatest element of its block.
OccurrencesOf1Over2Over3Openers3Closers12 -- Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 and 2 the greatest elements of their blocks.
OccurrencesOf1Over2Over3Openers3Closers2 -- Occurrences of the pattern 1/2/3 with 3 the least element of its block and 2 the greatest element of its block.
OpenerBeforeCloserPairs -- Pairs a < b in different blocks with a the least and b the greatest element of its block.
OverlappingBlockPairs -- Pairs of blocks A, B with min A < min B < max A < max B.
RightCloserBigger -- Pairs of a closer j and a smaller element i of an earlier block.
RightCloserSmaller -- Pairs of a closer j and a larger element i of an earlier block.
RightOpenerBigger -- Pairs of an opener j and a smaller element i of an earlier block.
RightOpenerSmaller -- Pairs of an opener j and a larger element i of an earlier block.
RotationOrbitSize -- The number of distinct partitions rotation i -> i + 1 (mod n) makes of this one.
SameBlockPairs -- Pairs of elements in one block.
SameBlockTriples -- Triples of elements in one block.
SetPartitionDimension -- The sum of the lengths of the arcs, taking for each block its largest element minus its smallest.
SetPartitionRank -- The number of elements minus the number of blocks.
SingletonBeforeCloserPairs -- Pairs a < b in different blocks with a a singleton block and b the greatest element of its block.
SingletonBeforeLargerPairs -- Pairs a < b in different blocks with a a singleton block.
SingletonBlocks -- Blocks containing exactly one element.
SingletonsAndAntisingletons -- Singleton blocks plus antisingletons.
SmallerBeforeCloserPairs -- Pairs a < b in different blocks with b the greatest element of its block.
SmallestBlock -- The size of the smallest block.
SmallestCloser -- The smallest of the blocks' largest elements.
SumOfBlockMaxima -- The sum of the largest elements of the blocks.
SumOfBlockMinima -- The sum of the smallest elements of the blocks.
TerminalClosers -- Closers e with every larger element also a closer.
TopologicallyConnectedComponents -- The topologically connected components of the arc diagram: blocks joined when an arc of one crosses an arc of the other.
UnsplittableFactorCount -- The number of unsplittable factors of the set partition under the split product.

Signed permutation statistics

FlagDenertIndex -- The negative entries plus twice the Denert index of the permutation |p(1)| ... |p(n)|.
FlagDescent -- Twice the descents between entries, plus 1 when the first entry is negative.
FlagExcedance -- Twice the excedances plus the negative entries.
FlagMajorIndex -- The number of negative entries plus twice the sum of the descents, a negative first entry counting as a descent at 0.
FlagWeakExcedance -- Twice the weak excedances plus the negative entries.
IncreasingPositivePairCount -- Pairs i < j with 0 < p(i) < p(j): the occurrences of the signed pattern 12.
NegativeEntryCount -- The number of negative entries.
NegativeSumPairCount -- Pairs i < j with p(i) + p(j) < 0.
PositiveBeforeLargerNegativePairCount -- Pairs i < j with 0 < p(i) < -p(j): the occurrences of the signed pattern 1-2.
PositiveBeforeNegativeCount -- Twice the adjacent pairs of a positive entry followed by a negative one, plus 1 when the first entry is negative.
PositiveEntryCount -- The number of positive entries.
RightDescents -- Indices 0 <= k < n with s(k) > s(k+1), where s is the inverse of p and s(0) = 0.
SignedAlignmentCount -- The alignments of p: its nesting alignments, those of type EN and those of type NE.
SignedPermutationOrder -- The least m >= 1 with p^m the identity, p acting on -n..n.
SignedReflectionLength -- The least number of reflections of the hyperoctahedral group with product p: n minus the positive cycles.
SignedSortingIndex -- The sum, over the signed transpositions that sort p by moving n, then n - 1, and so on into place, of j - i - 1 when i < 0 and j - i otherwise.
StasinskiVollLength -- Half the pairs -n <= i < j <= n of opposite parity with p(i) > p(j), where p(-k) = -p(k) and p(0) = 0.
TypeBInversionCount -- Pairs i < j with p(i) > p(j), plus pairs i <= j with -p(i) > p(j).

Skew partition statistics

FirstRowMissingBoxCount -- The boxes of the first row that the shape leaves out: the first part of the inner partition.
FrobeniusRank -- The fewest border strips that the shape divides into.
InnerCornerCount -- The cells that can be added to the inner partition, within the rows of the shape.
MissingBoxCount -- The boxes of the outer partition that the shape leaves out: the size of the inner partition.
SkewConnectedComponentCount -- The connected components of the shape: rows that start a component, the first row and every row that shares no column with the row above.
SkewCornerCount -- The cells that are an inner corner or an outer corner of the shape, counted once when both.
TwoByTwoSquareCount -- The cells of the shape whose neighbors to the right, below and below to the right are also in it.

Special functions 12 ≝37 primitive

BarnesG≝ -- The Barnes G-function $G(z)$ , the double-gamma function satisfying $G(z+1) = \Gamma(z)\,G(z)$ with $G(1) = 1$ — so $G(n) = \prod_{k=0}^{n-2} k!$ , the superfactorial, at positive integers.
BellYkernel -- The partial (incomplete) Bell polynomial $B_{n,k}(x_1, x_2, \dots)$ , by its textbook recurrence — built entirely from bigint binomial coefficients and compute-engine's own Add/Multiply, so it works the same for numeric or symbolic $x_i$ .
BesselJZeronumeric -- The $k$ -th positive zero of the Bessel function $J_\nu$ , for real $\nu > -1$ and positive integer $k$ . Provided by @enumeratio/analytic.
Beta -- Euler's beta function $B(a, b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$ , a bridge between Gamma and Binomial and the normalizing constant behind the beta distribution.
BetaRegularized -- The regularized incomplete beta function $I_x(a, b)$ , the CDF of the Beta(a, b) distribution and a bridge to the binomial CDF.
ChebyshevTkernel -- The Chebyshev polynomial of the first kind $T_n(x)$ , defined by $T_n(\cos\theta) = \cos(n\theta)$ . Provided by @enumeratio/analytic.
ChebyshevUkernel -- The Chebyshev polynomial of the second kind $U_n(x)$ , defined by $U_n(\cos\theta)\sin\theta = \sin((n+1)\theta)$ . Provided by @enumeratio/analytic.
ClausenCl≝ -- The Clausen functions $\mathrm{Cl}_n(\theta)$ : $\sum_{k\ge1} \sin(k\theta)/k^n$ for even $n$ and $\sum_{k\ge1} \cos(k\theta)/k^n$ for odd $n$ — the imaginary or real part of $\operatorname{Li}_n(e^{i\theta})$ .
DifferenceRoot -- A holonomic (P-recursive) sequence object — a linear recurrence with polynomial(n) coefficients plus enough initial values, exactly as DifferenceRootReduce emits it. Applied to a concrete non-negative integer $n_0$ , it runs the recurrence forward in exact rational arithmetic; applied to a symbolic $n$ (or anything else it can't run), it stays inert. Provided by @enumeratio/analytic.
DifferenceRootReduce -- Reduces a hypergeometric-type sequence $f(n)$ to its holonomic (P-recursive) representation — $\mathrm{DifferenceRoot}$ , a linear recurrence with polynomial(n) coefficients plus enough initial values to pin the sequence down, applied straight back to $n$ . Covers Factorial, $a^n$ , CatalanNumber, Binomial(n, k) for concrete k, polynomials in n, Fibonacci, LucasL, HarmonicNumber, and a sum of two hypergeometric terms; declines everything else, leaving the call symbolic. Provided by @enumeratio/analytic.
DifferentialRoot -- A holonomic (D-finite) function object — a linear ODE with polynomial(x) coefficients plus enough derivatives at one point, exactly as DifferentialRootReduce emits it. Applied to a real number, it sums the ODE's Taylor series about the stored expansion point numerically; applied to anything it can't sum (a symbolic point, or one outside the radius of convergence), it stays inert. Provided by @enumeratio/analytic.
DifferentialRootReduce -- Reduces a function $f(x)$ to its holonomic (D-finite) representation — $\mathrm{DifferentialRoot}$ , a linear ODE with polynomial(x) coefficients plus enough derivatives at one point, applied straight back to $x$ . Covers $e^{ax}$ , $\sin(ax)$ , $\cos(ax)$ , polynomials, a single-pole rational function, $\ln(1{+}x)$ , $\arctan(x)$ , $\mathrm{erf}(x)$ , BesselJ, AiryAi and AiryBi; declines sums, products, and anything else. Provided by @enumeratio/analytic.
Digamma -- The digamma function $\psi(z) = \dfrac{d}{dz}\ln\Gamma(z)$ , the logarithmic derivative of Gamma.
DigammaFunctionZeronumeric -- The $n$ -th real zero of the digamma function $\psi$ : $n=0$ names the zero on $(0,\infty)$ ( $x_0 \approx 1.4616$ ), $n \ge 1$ the zero on $(-n, -n+1)$ . Provided by @enumeratio/analytic.
DirichletBeta≝ -- The Dirichlet beta function $\beta(s) = \sum_{n\ge0} (-1)^n (2n+1)^{-s}$ , the L-function of the nontrivial character mod 4 — entire, with $\beta(1) = \pi/4$ and $\beta(2) = G$ (Catalan).
DirichletCharacterkernel -- The $j$ -th Dirichlet character modulo $k$ evaluated at $n$ : a completely multiplicative, $k$ -periodic map $\chi_j: \mathbb{Z} \to \mathbb{C}$ , zero where $\gcd(n, k) > 1$ .
DirichletEta≝ -- The Dirichlet eta function $\eta(s) = \sum_{n\ge1} (-1)^{n-1} n^{-s} = (1 - 2^{1-s})\,\zeta(s)$ , the alternating zeta — entire, with $\eta(1) = \ln 2$ .
DirichletL≝ -- The Dirichlet L-function $L(s, \chi) = \sum_{n\ge1} \chi(n)\,n^{-s}$ of the $j$ -th character mod $k$ — the family that contains $\zeta$ , $\eta$ and $\beta$ , and the setting of Dirichlet's theorem on primes in arithmetic progressions.
EisensteinG -- The un-normalized weight- $k$ Eisenstein series $G_k(\tau) = 2\zeta(k)\,E_k(\tau)$ over the upper half-plane, for even $k \geq 4$ .
EllipticTheta -- The Jacobi theta function $\vartheta_a(u,q)$ , $a=1,2,3,4$ (DLMF 20.2), Wolfram's nome convention $|q|<1$ .
EllipticThetaPrime -- The u-derivative of the Jacobi theta function, $\vartheta_a'(u,q) = \dfrac{\partial}{\partial u}\vartheta_a(u,q)$ , $a=1,2,3,4$ , Wolfram's nome convention $|q|<1$ .
Erf -- The Gauss error function $\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt$ , the shape behind the normal distribution's CDF.
ErfInv -- The inverse error function: $\operatorname{erfinv}(x)$ solves $\operatorname{erf}(y) = x$ for $-1 < x < 1$ .
Erfc -- The complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$ , the Gaussian's tail-probability integral.
ExpIntegralEkernel -- The generalized exponential integral $E_n(z) = \int_1^\infty e^{-zt}/t^n\,dt$ . Built on this package's generalized incomplete Gamma: $E_n(z) = z^{n-1}\,\Gamma(1-n, z)$ .
Gamma -- The gamma function $\Gamma(z)$ , extending the factorial to real and complex arguments; with two arguments, the upper incomplete gamma function.
GammaLn -- The natural logarithm of $\Gamma(z)$ , avoiding the overflow of computing Gamma directly for large z.
GammaRegularized -- The regularized upper incomplete gamma function $Q(a, z) = \Gamma(a, z)/\Gamma(a)$ , the survival function of the Gamma(a, 1) distribution.
HarmonicNumber≝ -- The harmonic number $H_n = \sum_{k=1}^n \tfrac1k$ , exact at a non-negative integer $n$ ; with a second argument, the generalized $H_n^{(r)} = \sum_{k=1}^n k^{-r}$ . Continued off the integers by $H_z = \psi(z+1) + \gamma$ . Provided by @enumeratio/analytic.
HermiteH≝ -- The physicists' Hermite polynomial $H_n(x)$ , orthogonal on the real line with weight $e^{-x^2}$ . Provided by @enumeratio/analytic.
HurwitzZeta≝ -- The Hurwitz zeta function $\zeta(s, a) = \sum_{n=0}^{\infty} (n+a)^{-s}$ for $\operatorname{Re}(s) > 1$ , continued analytically elsewhere. Provided by @enumeratio/analytic.
Hyperfactorialnumeric -- The hyperfactorial $H(n) = \prod_{k=1}^n k^k$ , continued to complex arguments: $H(z) = \Gamma(z+1)^z / G(z+1)$ .
Hypergeometric0F1numeric -- The confluent hypergeometric limit function ${}_0F_1(b; z) = \sum_{k\ge0} z^k / ((b)_k\,k!)$ — entire in $z$ , related to the Bessel functions. Provided by @enumeratio/analytic.
Hypergeometric0F1Regularizednumeric -- The regularized confluent hypergeometric limit function ${}_0F_1(b; z) / \Gamma(b)$ — entire in both $b$ and $z$ , unlike Hypergeometric0F1 itself. Provided by @enumeratio/analytic.
Hypergeometric1F1Regularizednumeric -- The regularized Kummer confluent hypergeometric function ${}_1F_1(a,b;z) / \Gamma(b)$ — entire in $b$ and $z$ . Provided by @enumeratio/analytic.
Hypergeometric2F1Regularizednumeric -- The regularized Gauss hypergeometric function ${}_2F_1(a,b,c;z) / \Gamma(c)$ (fungrim:fe6e74) — entire in $c$ ; only converges for $|z| < 1$ . Provided by @enumeratio/analytic.
Hypergeometric3F2Regularizednumeric -- The regularized generalized hypergeometric function ${}_3F_2(a_1,a_2,a_3;b_1,b_2;z) / (\Gamma(b_1)\Gamma(b_2))$ — entire in $b_1, b_2$ ; only converges for $|z| < 1$ . Provided by @enumeratio/analytic.
HypergeometricPFQkernel -- The generalized hypergeometric function ${}_pF_q(a; b; z)$ , built on the same series machinery (pfqSeries) as this package's 0F1/1F1Regularized/2F1Regularized/3F2Regularized, generalized to arbitrary $p$ and $q$ .
HypergeometricUnumeric -- Tricomi's confluent hypergeometric function $U(a,b,z)$ , for $b$ not an integer — see HypergeometricUStar for the $z^a$ -regularized form Fungrim builds most of its identities from. Provided by @enumeratio/analytic.
HypergeometricUStarnumeric -- The regularized Tricomi confluent hypergeometric function $U^*(a,b,z) = z^a\,U(a,b,z)$ , for $b$ not an integer. Provided by @enumeratio/analytic.
IncompleteEllipticPi -- The incomplete Legendre elliptic integral of the third kind, $\Pi(n;\varphi,m) = \int_0^\varphi \dfrac{d\theta}{(1-n\sin^2\theta)\sqrt{1-m\sin^2\theta}}$ , in Wolfram/mpmath's $(n,\varphi,m)$ order with $m=k^2$ .
InverseBetaRegularizedkernel -- The inverse of the regularized incomplete beta $I_x(a, b)$ in $x$ : the Beta-distribution quantile. Solved the same way as InverseGammaRegularized — a safeguarded Newton's method against native BetaRegularized.
InverseErfckernel -- The inverse complementary error function: solves $\operatorname{erfc}(y) = s$ for $0 < s < 2$ . Not simply ErfInv(1 - s) — that loses precision exactly in the tail ( $s \to 0$ ) InverseErfc exists for.
InverseGammaRegularizedkernel -- The inverse of the regularized incomplete gamma $Q(a, z)$ in $z$ : solves $s = Q(a, z)$ . Solved numerically by a safeguarded Newton's method against compute-engine's own GammaRegularized, using the Gamma density (in log space) as the derivative.
JacobiAmplitude -- The Jacobi amplitude $\operatorname{am}(u,m)$ — the angle $\varphi$ inverting $u = F(\varphi,m)$ (EllipticF) — in Wolfram/mpmath's $m=k^2$ convention.
JacobiCD -- The Jacobi elliptic function $\operatorname{cd}(u,m) = \operatorname{cn}(u,m)/\operatorname{dn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiCN -- The Jacobi elliptic function $\operatorname{cn}(u,m)$ , one of the three primary Jacobi elliptic functions, in Wolfram/mpmath's $m=k^2$ parameter convention.
JacobiCS -- The Jacobi elliptic function $\operatorname{cs}(u,m) = \operatorname{cn}(u,m)/\operatorname{sn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiDC -- The Jacobi elliptic function $\operatorname{dc}(u,m) = \operatorname{dn}(u,m)/\operatorname{cn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiDN -- The Jacobi elliptic function $\operatorname{dn}(u,m)$ , one of the three primary Jacobi elliptic functions, in Wolfram/mpmath's $m=k^2$ parameter convention.
JacobiDS -- The Jacobi elliptic function $\operatorname{ds}(u,m) = \operatorname{dn}(u,m)/\operatorname{sn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiNC -- The Jacobi elliptic function $\operatorname{nc}(u,m) = 1/\operatorname{cn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiND -- The Jacobi elliptic function $\operatorname{nd}(u,m) = 1/\operatorname{dn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiNS -- The Jacobi elliptic function $\operatorname{ns}(u,m) = 1/\operatorname{sn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiSC -- The Jacobi elliptic function $\operatorname{sc}(u,m) = \operatorname{sn}(u,m)/\operatorname{cn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiSD -- The Jacobi elliptic function $\operatorname{sd}(u,m) = \operatorname{sn}(u,m)/\operatorname{dn}(u,m)$ (Glaisher's notation), in Wolfram/mpmath's $m=k^2$ convention.
JacobiSN -- The Jacobi elliptic function $\operatorname{sn}(u,m)$ , one of the three primary Jacobi elliptic functions, in Wolfram/mpmath's $m=k^2$ parameter convention.
JacobiZN -- Wolfram's Jacobi zeta function $Z(u,m) = E(\operatorname{am}(u,m),m) - \dfrac{E(m)}{K(m)}u$ , in Wolfram/mpmath's $m=k^2$ convention.
KeiperLiLambda -- The n-th Keiper–Li coefficient $\lambda_n = \dfrac{1}{(n-1)!}\dfrac{d^n}{ds^n}\left[s^{n-1}\log\xi(s)\right]_{s=1}$ , whose nonnegativity for every n is equivalent to the Riemann hypothesis.
Khinchinnumeric -- Khinchin's constant $K_0 = 2.68545\ldots$ , the almost-sure geometric mean of continued-fraction terms.
KleinInvariantJkernel -- Klein's absolute invariant $J(\tau) = j(\tau)/1728$ , normalised so $J(i) = 1$ and $J(\rho) = 0$ at the elliptic points. A thin wrapper over this package's own ModularJ (the un-normalised $j$ ).
LaguerreL≝ -- The Laguerre polynomial $L_n(x)$ , and its generalization $L_n^{(a)}(x)$ with a parameter $a$ , orthogonal on $(0, \infty)$ with weight $x^a e^{-x}$ . Provided by @enumeratio/analytic.
LambertWkernel -- The Lambert $W$ function (Wolfram's ProductLog), the inverse of $w\,e^w$ . compute-engine's native LambertW already covers the principal ( $k=0$ ) and lower-real ( $k=-1$ ) branches numerically; extended here with exact values at algebraically nice points and every other integer branch.
LegendrePolynomialkernel -- The Legendre polynomial $P_n(x)$ , orthogonal on $[-1, 1]$ with unit weight and normalised by $P_n(1) = 1$ . Provided by @enumeratio/analytic.
LerchPhi≝ -- The Lerch transcendent $\Phi(z, s, a) = \sum_{n=0}^{\infty} \dfrac{z^n}{(n+a)^s}$ — a common generalization of the polylogarithm and the Hurwitz zeta. Provided by @enumeratio/analytic.
LogBarnesG≝ -- The logarithm of the Barnes G-function, $\ln G(z)$ , as an analytic continuation — the form that stays finite where $G$ itself overflows.
LogGamma≝ -- The log-gamma function $\ln\Gamma(z)$ as an analytic continuation (branch cut on $(-\infty, 0]$ ), which differs from $\ln(\Gamma(z))$ by multiples of $2\pi i$ off the positive axis.
MeijerGkernel -- The Meijer G-function $G^{m,n}_{p,q}$ , evaluated numerically via its reduction to a finite sum of ordinary ${}_pF_q$ series (DLMF 16.17.2), run in BigDecimal (meijer-g-big.ts) so the reduction's own cancellation lands a few ulps of the true value rather than ~1e-13.
MeijerGReducekernel -- Rewrites a handful of elementary and special functions into MeijerG form, using identities that hold for the whole operand (not just a linear argument), so each is unconditional.
ModularJ -- Klein's modular function $j(\tau) = E_4(\tau)^3/\Delta(\tau)$ over the upper half-plane, invariant under the full modular group $\mathrm{SL}_2(\mathbb{Z})$ .
MultiZetaValuenumeric -- The depth-2 multiple zeta value $\zeta(s_1, s_2) = \sum_{n_1 > n_2 \ge 1} n_1^{-s_1} n_2^{-s_2}$ , for integers $s_1, s_2 \ge 2$ . Provided by @enumeratio/analytic.
NorlundBkernel -- The Nörlund polynomial $B_n^{(a)}$ , from the generating function $(t/(e^t-1))^a$ . At $a=1$ it is the ordinary Bernoulli number; in general, an exact polynomial in $a$ with bigint-rational coefficients, from a power-series log/exp of the Bernoulli EGF.
PolyGamma -- The polygamma function $\psi^{(m)}(z) = \dfrac{d^{m+1}}{dz^{m+1}} \ln\Gamma(z)$ , the $m$ -th derivative of the digamma.
PolyLog -- The polylogarithm $\operatorname{Li}_s(z) = \sum_{n=1}^{\infty} \dfrac{z^n}{n^s}$ , the $a = 1$ slice of the Lerch transcendent.
PrimeZetaPkernel -- The prime zeta function $P(s) = \sum_p p^{-s}$ , the sum over primes. Computed via the Möbius/ζ identity $P(s) = \sum_{k \ge 1} \mu(k)/k \cdot \ln\zeta(ks)$ rather than sieving primes directly — the identity converges geometrically, sieving does not.
QBinomialkernel -- The Gaussian (q-)binomial coefficient $\binom{n}{k}_q$ , the generating polynomial in $q$ for partitions fitting in a $k \times (n-k)$ box. Provided by @enumeratio/analytic.
QFactorialkernel -- The q-factorial $[n]_q! = [1]_q [2]_q \cdots [n]_q$ , where $[k]_q = 1 + q + \cdots + q^{k-1}$ . Provided by @enumeratio/analytic.
QPochhammerkernel -- The q-Pochhammer symbol $(a; q)_n = \prod_{k=0}^{n-1} (1 - a q^k)$ , for a nonnegative integer $n$ or $n = \infty$ . Provided by @enumeratio/analytic.
RiemannSiegelThetanumeric -- The Riemann–Siegel theta function $\vartheta(t) = \operatorname{Im}\ln\Gamma(\tfrac14 + \tfrac{it}{2}) - \tfrac{t}{2}\ln\pi$ , for real $t$ . Provided by @enumeratio/analytic.
RiemannSiegelZnumeric -- The Riemann–Siegel function $Z(t) = e^{i\vartheta(t)}\zeta(\tfrac12 + it)$ , real-valued for real $t$ by construction. Provided by @enumeratio/analytic.
RiemannZetaZeronumeric -- The $k$ -th nontrivial zero of the Riemann zeta function on the critical line, $\tfrac12 + i t_k$ for a positive integer $k$ . Wolfram calls this ZetaZero; compute-engine's own Fungrim identity table already spells it RiemannZetaZero. Provided by @enumeratio/analytic.
SloaneAnumeric -- The $n$ -th term of an OEIS sequence, id a quoted A-number — scoped to the specific sequences Fungrim's own SloaneA identities cite, each aliased to a head that already computes it. Provided by @enumeratio/analytic.
StieltjesGammanumeric -- The Stieltjes constants $\gamma_n$ , the coefficients of the Laurent expansion of $\zeta(s)$ at $s = 1$ : $\zeta(s) = \dfrac{1}{s-1} + \sum_{n\ge0} \dfrac{(-1)^n}{n!}\gamma_n (s-1)^n$ ; with a second argument, the generalized $\gamma_n(a)$ for HurwitzZeta.
Zetanumeric -- The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ for $\operatorname{Re}(s) > 1$ , continued analytically elsewhere.

Statistics

BernoulliDistribution -- The Bernoulli distribution: $1$ with probability $p$ , $0$ with probability $1-p$ .
BetaDistribution -- The Beta distribution on $[0, 1]$ with shape parameters $\alpha, \beta$ .
BinormalDistribution -- The bivariate normal distribution over pairs $(x_1, x_2)$ .
CauchyDistribution -- The Cauchy distribution with location $a$ and scale $b$ — heavy-tailed enough that its mean and variance don't exist.
CentralMoment -- The central moment $E[(X-Mean)^r]$ .
CharacteristicFunction -- The characteristic function $\varphi(t) = E[e^{itX}]$ of $dist$ .
ChiDistribution -- The chi distribution with $k$ degrees of freedom — the distribution of $\sqrt{\chi^2_k}$ .
ChiSquareDistribution -- The chi-square distribution with $k$ degrees of freedom — $GammaDistribution(k/2, 2)$ under the hood.
Conditioned -- $a$ conditioned on $b$ — meaningful only as Probability's or Expectation's first argument.
Cumulant -- The cumulant $\kappa_r$ .
DirichletDistribution -- The Dirichlet distribution on the $(k-1)$ -simplex with concentration parameters $\alpha_1,\ldots,\alpha_k$ .
DiscreteUniformDistribution -- The discrete uniform distribution on the integers $\{min, \ldots, max\}$ .
Distributed -- Binds a variable to a distribution, for Expectation and Probability.
EmpiricalDistribution -- The distribution of the values actually observed in data.
ErlangDistribution -- The Erlang distribution with shape $n$ and rate $\lambda$ — $GammaDistribution(n, 1/\lambda)$ under the hood.
Expectation -- $E[f(x)]$ — the expected value of f under x's distribution.
FactorialMoment -- The factorial moment $E[X(X-1)\cdots(X-r+1)]$ .
GammaDistribution -- The Gamma distribution with shape $k$ and scale $\theta$ .
GeometricDistribution -- The Geometric distribution: the number of failures before the first success, with success probability $p$ .
HalfNormalDistribution -- The half-normal distribution with parameter $\theta$ — $\sqrt{\pi/2}\,|Z|/\theta$ for a standard normal $Z$ .
HazardFunction -- The hazard function $PDF(dist, x) / SurvivalFunction(dist, x)$ .
HistogramDistribution -- The empirical density of $data$ , binned into equal-width bins of width $dx$ — piecewise-constant.
HypergeometricDistribution -- The hypergeometric distribution: $n$ draws without replacement from a population of $ntotal$ with $nsucc$ successes.
InverseCDF -- The quantile function: the $x$ such that $CDF(dist, x) = q$ .
LaplaceDistribution -- The Laplace (double-exponential) distribution with location $\mu$ and scale $b$ .
LogNormalDistribution -- The log-normal distribution: $\log X \sim NormalDistribution(\mu, \sigma)$ .
LogisticDistribution -- The logistic distribution with location $\mu$ and scale $\beta$ .
MarginalDistribution -- The $k$ -th marginal factor of a ProductDistribution — or the joint marginal of several.
MaxwellDistribution -- The Maxwell(-Boltzmann) distribution with scale $\sigma$ — $\sigma$ times a ChiDistribution( $3$ ) draw.
MixtureDistribution -- The weighted mixture of $d_1, \ldots, d_n$ with (renormalized) weights $w_1, \ldots, w_n$ .
Moment -- The raw moment $E[X^r]$ .
MomentGeneratingFunction -- The moment generating function $M(t) = E[e^{tX}]$ of $dist$ .
MultinomialDistribution -- $n$ draws over $k$ categories with probabilities $p_1,\ldots,p_k$ (summing to 1); a count vector on the $n$ -simplex.
MultinormalDistribution -- The multivariate normal with mean vector $\mu$ and covariance matrix $\Sigma$ ; the one-argument form defaults $\mu$ to the zero vector.
MultivariatePoissonDistribution -- Wolfram's common-shock model — $X_i = Y_0 + Y_i$ , a shared $Y_0 \sim Poisson(\mu_0)$ plus an independent $Y_i \sim Poisson(\mu_i)$ per coordinate.
NExpectation -- A numeric-only $E[f(x)]$ — falls back to summation/quadrature where Expectation stays symbolic.
NProbability -- A numeric-only $P(cond)$ — handles any condition compute-engine can evaluate at a point, not just the shapes Probability recognizes symbolically.
NegativeBinomialDistribution -- The negative binomial distribution: the number of failures before the $n$ -th success, with success probability $p$ .
ParameterMixtureDistribution -- A distribution whose own parameter $\theta$ is itself random, drawn from $prior$ — the marginal of $dist(\theta)$ averaged over $\theta \sim prior$ .
ParetoDistribution -- The Pareto distribution with minimum value $k$ and shape $\alpha$ .
PoissonProcess -- The counting process of events arriving at rate $\lambda$ per unit time.
Probability -- $P(cond)$ under x's distribution.
ProbabilityDistribution -- A distribution given directly by its density $pdf$ , over $x \in [min,max]$ (continuous) or $x \in \{min,\ldots,max\}$ (discrete, step size 1).
ProductDistribution -- The joint distribution of $n$ INDEPENDENT $d_1, \ldots, d_n$ — an $n$ -dimensional carrier.
RandomFunction -- One seeded sample path of a random process over a time grid.
RandomVariate -- A random draw from a distribution, or a list of them -- Random over the distribution, by its Wolfram name.
RayleighDistribution -- The Rayleigh distribution with scale $\sigma$ — the distribution of the magnitude of a 2D Gaussian vector.
SliceDistribution -- The ordinary distribution answering a random process's value at a fixed time $t$ .
StudentTDistribution -- The Student-t distribution with $\nu$ degrees of freedom.
SurvivalFunction -- The survival function $1 - CDF(dist, x)$ — $P(X > x)$ .
TransformedDistribution -- The distribution of $expr(x)$ where $x \sim dist$ — exact for $expr$ affine in $x$ , plus $x^2$ of a standard normal.
TriangularDistribution -- The triangular distribution on $[a,b]$ with mode $c$ (defaulting to the midpoint).
TruncatedDistribution -- $dist$ restricted to $[a,b]$ (continuous) or $(a,b]$ (discrete), renormalized.
WeibullDistribution -- The Weibull distribution with shape $\alpha$ and scale $\beta$ .
WienerProcess -- Standard Brownian motion with drift $\mu$ and volatility $\sigma$ , starting at $0$ .

Structures

AlgebraDimension -- The dimension of a finite-dimensional algebra: the length of its basis.
Basis -- The basis of a finite-dimensional algebra, as a list.
CombinatorialMap -- A combinatorial map applied to a value, by name or FindStat id, found through the value's carrier.
CombinatorialStat -- A combinatorial statistic of a value, by name or FindStat id, found through the value's carrier.
Compare -- How two values of a partially ordered type compare: -1, 0 or 1, or NaN when they are incomparable.
Coordinates -- The coordinates of a value of a product order, as a list.
GeometricProduct -- Alias of NonCommutativeMultiply, for the geometric-algebra reading of the ordered product.
GreatestLowerBound -- The meet of two values in a lattice: the greatest value at or below both.
HasElement -- Whether x is an element of a finite-dimensional algebra.
IntegerCeil -- The least integer at or above x, in a floor ring.
IntegerFloor -- The greatest integer at or below x, in a floor ring.
IsEvenTick -- Whether a tick is an even one, in a type whose ticks have a parity.
LeastUpperBound -- The join of two values in a lattice: the least value at or above both.
LowerTick -- The greatest tick at or below x, in a type whose order has ticks.
Midpoint -- The point halfway between two values.
NonCommutativeMultiply -- The ordered product, for units that anticommute. $\times$ cannot host one: Multiply is declared commutative, so canonicalisation sorts its operands before any handler runs and the sign is lost.
OrderOfElement -- The order of a group element, the least n ≥ 1 with $g^n$ the identity, or Infinity when there is none.
UpperTick -- The least tick at or above x, in a type whose order has ticks.
WithCoordinates -- A value shaped like x with the given coordinates.

Tableau statistics

AdjacentSuccessorCount -- Entries whose successor is the neighbor to their right or directly below them.
AscentCountIncludingLargestEntry -- Entries whose successor is in the same or a higher row, and the largest entry.
AttackingInversionCount -- Attacking pairs of cells whose entries are out of order: the cell in the lower row, strictly east of the other, holds the larger entry.
AttackingPairCount -- Pairs of cells of the shape with one strictly west of the other in a row, or the first in the row below and strictly east of the second.
Charge -- The charge of the reading word: the sum of n - e over the entries e whose successor is not in a lower row.
Cocharge -- The cocharge of the reading word: the sum of n - e over the entries e whose successor is in a lower row.
ComajorIndexMinusWeightedSize -- The sum of n - e over the descents e of the tableau, less the weighted size of its shape.
CylindricalTableauOverlap -- The number of rows, from the top, as long as the first row and starting, from the bottom, above the last entry of the first row.
FirstColumnInitialSequenceLength -- The largest k such that the first column starts 1, 2, ..., k.
FirstEntryInLastRow -- The first entry in the last row.
HaglundStevensInversionCount -- The pairs of a cell's entry and a smaller entry lying below the cell's inversion path, over all cells.
LastEntryInFirstRow -- The last entry in the first row.
LastEntryOnMainDiagonal -- The last entry on the main diagonal.
MajorIndexMinusWeightedSize -- The major index of the tableau less the weighted size of its shape.
MinimalCylindricalInnerHeight -- The number of rows less the overlap of the cylindrical tableau.
ModularMajorIndex -- The major index modulo the number of entries.
NaturalComajorIndex -- The sum of n - e over the natural descents e, the entries whose successor is in a strictly higher row.
NaturalDescentCount -- Entries whose successor is in a strictly higher row.
NaturalMajorIndex -- The sum of the natural descents, the entries whose successor is in a strictly higher row.
PathLengthToLargestEntry -- The length of the path from the top left cell to the largest entry, one step per cell down or to the right.
RowOfLargestEntry -- The row containing the largest entry.
ShiftedNaturalComajorIndex -- The sum of n - e over the natural descents e, plus the sum of (r - 1) times the length of row r.
ShynarInversionCount -- Pairs of entries e < f with f strictly southwest of e.
SmallestCylindricShift -- The least shift s at which each entry of the last row is smaller than the first row's entry s columns on.

The modular group

ContinuedFraction -- The regular continued fraction $[a_0; a_1, a_2, \dots]$ of $p/q$ . Its partial quotients are the run lengths of the rational's Stern–Brocot path, and the $T$ -exponents of its matrix — one object under three names.
ContinuedFractionK -- The continued fraction $f_1/(g_1 + f_2/(g_2 + \cdots))$ over an index range, finite or infinite.
Convergents -- The successive convergents $p_k/q_k$ of a continued fraction, given as its list of terms or as the number itself.
DedekindSum -- The Dedekind sum $s(h,k) = \sum_{j=1}^{k-1}((j/k))\,((hj/k))$ , returned as an exact rational rather than a float.
EvaluateForm -- The value of a QuadraticForm $ax^2+bxy+cy^2$ at the point $(x,y)$ .
FareyNeighbours -- Whether $p/q$ and $r/s$ are Farey neighbours — adjacent in some Farey sequence, which is the determinant test $|ps-qr|=1$ .
FormAction -- The image of a QuadraticForm under the modular group's action: substituting $(x,y)\mapsto(px+qy,rx+sy)$ for a matrix $\begin{pmatrix}p&q\\r&s\end{pmatrix}$ . The discriminant is invariant, which is why classes live inside one discriminant.
FormAutomorph -- The automorph: the hyperbolic matrix generating an indefinite QuadraticForm's stabiliser in $\mathrm{SL}(2,\mathbb{Z})$ , built from the fundamental PellSolution of its discriminant.
FormClassNumber -- How many classes of indefinite binary quadratic forms a discriminant has. A class is not one reduced form but a CYCLE of them, and that cycle is a periodic continued fraction — which is to say, a closed geodesic.
FormClasses -- Every class of indefinite forms of discriminant $D$ : its reduced forms, partitioned into $\rho$ -cycles.
FormCycle -- The cycle of reduced forms containing a QuadraticForm — the class it lies in, listed. Its length is always even.
FormDiscriminant -- The discriminant $D = b^2 - 4ac$ of a QuadraticForm.
FormRho -- One step round the reduction cycle: Gauss's $\rho$ operator on an indefinite QuadraticForm — the continued-fraction step, in form coordinates.
FromSternBrocotPath -- The positive rational at the end of a Stern–Brocot path of $L$ s and $R$ s — the inverse of SternBrocotPath.
IsIndefinite -- Whether a QuadraticForm is indefinite: its discriminant is positive and not a perfect square.
IsPrimitiveClass -- Whether an $LR$ word is primitive: not a proper power of a shorter word, so its conjugacy class is not a repeat of a shorter geodesic.
IsQuadraticIrrational -- True when $x$ is an irrational root of a quadratic with integer coefficients — exactly the numbers with an eventually periodic continued fraction.
IsReducedForm -- Whether an indefinite QuadraticForm satisfies Gauss's reduction condition, $|\sqrt D - 2|a|| < b < \sqrt D$ .
LinkingWithTrefoil -- The linking number of a hyperbolic element's modular knot with the trefoil — Ghys's theorem, and the same number as RademacherSymbol.
ModularClass -- The canonical name of a conjugacy class: the lexicographically least rotation of its $LR$ word.
ModularClasses -- Every hyperbolic conjugacy class whose $LR$ word has the given length — equivalently, every closed geodesic of that symbolic period on the modular surface. Conjugation rotates the word, so the classes are binary NECKLACES.
ModularFromSTWord -- Rebuild a matrix from its $S/T$ factorisation — the inverse of ModularSTWord.
ModularKind -- The trichotomy of an element by $|\mathrm{tr}|$ against 2: Identity, Elliptic, Parabolic or Hyperbolic.
ModularMatrix -- An element of $\mathrm{PSL}(2,\mathbb{Z})$ — or an $LR$ word standing for one. Wolfram has no dedicated modular-group heads: elements multiply, invert and raise to a power exactly like any other integer matrix, via Dot, Inverse and MatrixPower — which is why those are widened in place to recognise a ModularMatrix or a word, rather than declared afresh.
ModularSTWord -- The alternating $S/T$ factorisation of a matrix, $T^{e_0}ST^{e_1}S\cdots$ , as its list of $T$ -exponents — the continued fraction of the matrix, up to sign.
ModularTrace -- The trace of a matrix in $\mathrm{PSL}(2,\mathbb{Z})$ .
ModularWord -- The unique positive word in $L$ and $R$ of a matrix in $\mathrm{PSL}(2,\mathbb{Z})$ with non-negative entries — which is also its path down the Stern–Brocot tree.
PellSolution -- The fundamental solution $(t,u)$ , $u>0$ , of Pell's equation $t^2-Du^2=4$ .
QuadraticForm -- An indefinite binary quadratic form $ax^2+bxy+cy^2$ , carried by its coefficient triple.
RademacherPhi -- Rademacher's $\Phi$ function, built from Dedekind sums — the correction term that makes $\log\eta$ transform correctly under the modular group. Unlike RademacherSymbol it is not a class function.
RademacherSymbol -- The Rademacher symbol $\Psi$ of a hyperbolic element. By Ghys's theorem it is the linking number of the element's modular knot with the trefoil — and it is also just the number of $R$ 's minus the number of $L$ 's in its word.
ReduceForm -- Reduce an indefinite QuadraticForm by walking FormRho until Gauss's reduction condition holds.
ReducedForms -- Every reduced form of a given indefinite discriminant $D$ .
SternBrocotPath -- The Stern–Brocot path of a positive rational $p/q$ : its sequence of left and right turns down the tree from $1/1$ .
WordSymbol -- The Rademacher symbol read straight off an $LR$ word: the number of $R$ 's minus the number of $L$ 's.

Transformations 1 primitive

ComplexExpand -- Split expr into real and imaginary parts, treating every free symbol as real. Provided by @enumeratio/analytic.
ExpToTrig -- Rewrite every exponential in expr as circular or hyperbolic functions -- the inverse of TrigToExp. Provided by @enumeratio/analytic.
FullSimplify -- Simplify expr harder than Simplify, trying a wider set of transformations including special-function identities. Provided by @enumeratio/analytic.
FunctionExpand -- Rewrite special functions in expr in terms of more elementary or better-known ones. Provided by @enumeratio/analytic.
PowerExpand -- Expand powers and logarithms of products in expr as though every variable were positive. Provided by @enumeratio/analytic.
TrigFactorkernel -- The factoring counterpart of TrigExpand/TrigReduce — rewrites sums and differences of Sin/Cos into products.

Transforms 11 primitive

Fourierkernel -- The numeric discrete Fourier transform of a list, or of a rectangular matrix (a separable 2D transform), under Wolfram's default $\mathrm{FourierParameters} \to \{0, 1\}$ — $f_s = \frac{1}{\sqrt n}\sum_r \mathrm{list}_r\, e^{2\pi i (r-1)(s-1)/n}$ .
FourierCoefficientkernel -- The $n$ -th complex Fourier coefficient of $f$ on $[-\pi, \pi]$ , $c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\, e^{-inx}\,dx$ , for a concrete integer $n$ — a curated set of closed forms, not compute-engine's Integrate.
FourierSerieskernel -- The order- $n$ complex exponential Fourier series of $f$ on $[-\pi, \pi]$ , $\sum_{k=-n}^{n} c_k\, e^{ikx}$ , each $c_k$ from FourierCoefficient.
FourierTransformkernel -- The Fourier transform under Wolfram's default convention ( $\mathrm{FourierParameters} \to \{0, 1\}$ ) — $F(w) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{iwt}\,dt$ — as a rule table over the standard pairs plus linearity and the modulation (shift) theorem.
HankelTransformkernel -- The order- $n$ Hankel transform $F(s) = \int_0^\infty f(r)\,J_n(sr)\,r\,dr$ ( $n=0$ by default), as a rule table over the standard pairs.
InverseFourierkernel -- The numeric inverse discrete Fourier transform — Fourier with the FourierParameters sign flipped, $\{a, b\} \to \{-a, -b\}$ (confirmed against wolframscript), so it shares that head's kernel exactly.
InverseFourierTransformkernel -- The inverse Fourier transform under Wolfram's default convention — the two closed forms it documents unconditionally ( $\delta(w) \to 1/\sqrt{2\pi}$ and a constant $\to$ a scaled $\delta(t)$ ), nothing more.
InverseLaplaceTransformkernel -- The inverse Laplace transform, as a small dictionary of common images ( $s^{-n}$ , $1/(s-a)$ , $s/(s^2 \pm a^2)$ , $a/(s^2 \pm a^2)$ ) plus linearity — not a general residue calculus.
InverseMellinTransformkernel -- The inverse Mellin transform — $f$ such that $\mathcal{M}\{f\}(s) = F(s)$ — as a rule table mirroring MellinTransform's nine pairs.
LaplaceTransformkernel -- The one-sided Laplace transform $F(s) = \int_0^\infty f(t) e^{-st}\,dt$ , as a rule table over the standard pairs (powers, exponentials, trig/hyperbolic, the unit step, the impulse) plus linearity and the first shifting theorem.
MellinTransformkernel -- The Mellin transform $F(s) = \int_0^\infty f(x)\,x^{s-1}\,dx$ , as a rule table over nine standard pairs plus the scaling and power-shift theorems.

Tree statistics

AutomorphismGroupSize -- The number of ways to permute identical children, at every vertex at once: the product, over the vertices, of the factorials of how many times each child tree occurs.
BinaryNodes -- The number of nodes with two children.
BoundaryLength -- The edges on the leftmost path from the root plus those on the rightmost.
BoundaryLengthBeyondTrunk -- The edges on the leftmost path or the rightmost path from the root, less the trunk they share.
BoundaryNodes -- The number of nodes on the left spine or the right spine, the root counted once.
Breadth -- The most vertices at any one depth.
CollessIndex -- The sum, over the nodes, of how much the node counts of the two subtrees differ.
DistinctSubtrees -- BinaryTree: The number of different nonempty subtrees. OrderedTree: The number of different subtrees.
ExternalPathLength -- The sum of the lengths of the paths from the root to the empty subtrees.
ForkRightRightLeftPatternCount -- The number of nodes with two children whose right child's right child has a left child.
HookNumber -- The number of hooks in the decomposition of the tree into hooks: a node with the chains of left children and of right children below it, the subtrees hanging off those chains decomposed in turn.
InequivalentIncreasingTreeCount -- The number of increasing labelings of the tree up to its automorphisms: n! over the tree factorial and the automorphism group size.
LargestOutDegree -- The most children any vertex has.
Leaves -- The number of leaves, the vertices with no children.
LeftSpine -- The number of nodes on the left spine of a binary tree: the root and its chain of left children.
LeftSubtreeSize -- The number of nodes in the left subtree of the root.
LinearExtensionCount -- BinaryTree: The number of linear extensions of the tree as a poset: n! over the tree factorial. OrderedTree: The number of linear extensions of the tree as a poset: n! over the tree factorial.
LongestPathVertexCount -- The most vertices on a path from the root: the height plus one.
MaximalLeftBranches -- The number of maximal paths of left edges.
NonBoundaryNodes -- The number of nodes on neither the left spine nor the right spine.
NonRootBinaryNodes -- The number of nodes other than the root with two children.
NonRootInternalNodes -- The number of vertices that are neither the root nor a leaf.
OneProtectedNodeCount -- The number of vertices whose nearest leaf is exactly one edge away.
ParenthesisCodeValue -- The value, read in binary, of the tree's parenthesis word: each node is ( left ) right, with ( as 1 and ) as 0.
PathLength -- BinaryTree: The sum of the depths of the nodes, the root's being 0: the internal path length. OrderedTree: The sum of the depths of the vertices, the root's being 0.
Pitchforks -- The number of nodes whose subtree has exactly two nodes.
ProtectionNumber -- BinaryTree: The edges from the root to the nearest leaf, a binary tree's empty subtree counting as a leaf. OrderedTree: The edges from the root to the nearest leaf.
PruningNumber -- The number of prunings, each deleting every maximal hanging branch (a path of single-child vertices to a leaf), that reduce the tree to a single vertex.
RightChildCount -- The number of nodes with a right child.
RightForkRightLeftPatternCount -- The number of nodes whose right child has two children, the right one of which has a left child.
RightLeftLeftLeftPathCount -- The number of nodes from which taking the right, then the left, then the left, then the left child stays inside the tree.
RightLeftLeftPathCount -- The number of nodes from which taking the right, then the left, then the left child stays inside the tree.
RightRightLeftLeftPathCount -- The number of nodes from which taking the right, then the right, then the left, then the left child stays inside the tree.
RightRightLeftPathCount -- The number of nodes from which taking the right, then the right, then the left child stays inside the tree.
RightRightLeftRightPathCount -- The number of nodes from which taking the right, then the right, then the left, then the right child stays inside the tree.
RightRightPathCount -- The number of nodes from which taking the right, then the right child stays inside the tree.
RightRightRightLeftPathCount -- The number of nodes from which taking the right, then the right, then the right, then the left child stays inside the tree.
RightRightRightPathCount -- The number of nodes from which taking the right, then the right, then the right child stays inside the tree.
RightRightRightRightPathCount -- The number of nodes from which taking the right, then the right, then the right, then the right child stays inside the tree.
RightSubtreeSizeSum -- The sum, over the nodes, of the number of nodes in the right subtree.
RootDegree -- The number of children of the root.
StrahlerNumber -- BinaryTree: The Horton-Strahler number (register function): 0 for the empty tree, else the larger child's, plus one if the children tie. OrderedTree: The Strahler number: a leaf is 1, a vertex takes its largest child's, plus one if that value occurs twice.
SwapEquivalentTreeCount -- The number of binary trees that become this one when the children at any nodes are swapped: each node with different subtrees (an only child counts as different from the empty one) doubles the count.
SwapEquivalentTreeCountLog -- The binary logarithm of the number of binary trees with the same underlying rooted tree: the nodes whose two subtrees differ up to swaps.
TreeFactorial -- BinaryTree: The product, over the nodes, of the number of nodes at or below each. OrderedTree: The product, over the vertices, of the number of vertices at or below each.
TrunkLength -- The edges from the root to the first vertex that does not have exactly one child.
TwoProtectedNodeCount -- The number of vertices whose nearest leaf is exactly two edges away.
UnaryNodes -- The number of nodes with exactly one child.
VertexHeightSum -- The sum, over the nodes, of the node's height (the most nodes on a path down).

Wolfram

BernoulliPolynomial -- The Bernoulli polynomial $B_n(x)$ -- Wolfram's own BernoulliB(n, x).
Cell -- Wolfram's own Cell, mapped through for the transpiler and the oracle but not yet written up here.
ConstGlaisher -- The Glaisher-Kinkelin constant $A \approx 1.2824271291006226\ldots$ -- Wolfram's own Glaisher.
FallingFactorial -- The falling factorial $x(x-1)\cdots(x-n+1)$ -- Wolfram's own FactorialPower(x, n).
Notebook -- Wolfram's own Notebook, mapped through for the transpiler and the oracle but not yet written up here.
Rasterize -- Wolfram's own Rasterize, mapped through for the transpiler and the oracle but not yet written up here.

Collections

7 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Combinatorics

177 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Compute engine

537 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Number theory

1 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

QuadraticInteger

Oracle

11 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Wolfram

21 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Carrier domains

9 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Symbol reference ​

Elsewhere ​

Adèles and idèles

Alternating sign matrix statistics

Analytic 2 primitive

Arithmetic

Binary word statistics

Boxes

Braids and knots

Calculus

Catalog

Collections

Combinatorial Hopf algebras

Combinatorial collections

Combinatorial maps

Combinatorics 1 ≝

Composition statistics

Compute engine

Controlling evaluation

Diagram algebras

Dyck path and binary word statistics

Dyck path statistics

Elementary functions 1 primitive

Geometric algebra

Graph statistics

Graphics

Group algebras

Hecke algebras

Hopf algebras

Hypercomplex algebra

Incidence algebras

Interval arithmetic

Linear algebra

Modular arithmetic

Number theory 8 primitive

Numeral systems

Parking function statistics

Partition and standard tableau statistics

Partition statistics

Path algebras

Path and tree statistics

Perfect matching statistics

Permutation statistics 1 ≝1 primitive

Permutations

Plane partition statistics

Relations

Sequences

Set partition statistics

Signed permutation statistics

Skew partition statistics

Special functions 12 ≝37 primitive

Statistics

Structures

Tableau statistics

The modular group

Transformations 1 primitive

Transforms 11 primitive

Tree statistics

Wolfram

Collections

Combinatorics

Compute engine

Number theory

Oracle

Wolfram

Carrier domains

Symbol reference

Elsewhere