IntegerPartitions

The partitions of $n$ into positive parts, as a lazy indexed family.

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IntegerPartitions(n)every way to write

n

as a sum of positive parts, order-insensitive.

CombinatoricaPartitions[n]

IntegerPartitions(n, k)the partitions of

n

into at most

k

parts.

Combinatorica

If[IntegerQ[k], Reverse[Sort[Map[TransposePartition, Partitions[n, k]], Function[{a, b}, OrderedQ[{PadRight[a, n], PadRight[b, n]}]]]], If[ListQ[k] && 1 <= Length[k] <= 3, Select[Partitions[n], MemberQ[Switch[Length[k], 1, k, 2, Range[k[[1]], Min[k[[2]], n]], 3, Range[k[[1]], Min[k[[2]], n], k[[3]]]], Length[#]] &], Missing["NotCombinatorica"]]]

IntegerPartitions(n, {k})the partitions of

n

into exactly

k

parts.

Combinatorica

If[IntegerQ[{k}], Reverse[Sort[Map[TransposePartition, Partitions[n, {k}]], Function[{a, b}, OrderedQ[{PadRight[a, n], PadRight[b, n]}]]]], If[ListQ[{k}] && 1 <= Length[{k}] <= 3, Select[Partitions[n], MemberQ[Switch[Length[{k}], 1, {k}, 2, Range[{k}[[1]], Min[{k}[[2]], n]], 3, Range[{k}[[1]], Min[{k}[[2]], n], {k}[[3]]]], Length[#]] &], Missing["NotCombinatorica"]]]

IntegerPartitions(n, All, parts)the partitions of

n

using only parts drawn from the given list.

Typed signatures

Type	Package	Applies
`(integer, any?, any?) -> indexed_collection<integer_partition>`	`combinatorics`	always

Domain: Collections

Details

A lazy indexed collection, unranked in reverse-lexicographic order; the count is the partition number $p(n)$ — $p(8) = 22$ — with no elementary closed form.
Each element is the part list in weakly decreasing order; drawn as a Ferrers diagram, its statistics (Length, LargestPart, DurfeeSquare, …) live in $@enumeratio/statistics$ .
Conjugation (transposing the diagram) is an involution; the self-conjugate partitions of $n$ equal the partitions of $n$ into distinct odd parts.

Definition

Epsil, as the implementation runs it.

count(_n)

_tables[_n * (_n + 1) + _n + 1]

unrank(_n, _r)

Drop(Fold(Function(
                    Apply((pw_m: integer) => Apply((pw_c: integer) => Apply((pw_r: integer) => pw_u if pw_m == 0 else Apply(
                                                                                                                             (pw_chosen: list<integer>) => Join(
                                                                                                                                                                 [pw_m - pw_chosen[2], pw_chosen[2], pw_chosen[1]],
                                                                                                                                                                 Drop(pw_u, 3),
                                                                                                                                                                 [pw_chosen[2]],
                                                                                                                                                               ),
                                                                                                                             Fold(
                                                                                                                                   Function(pw_pick if pw_pick[2] != 0 else Apply(
                                                                                                                                                                                   (pw_k: integer) => [pw_pick[1], pw_p] if pw_pick[1] < pw_k else [pw_pick[1] - pw_k, 0],
                                                                                                                                                                                   _tables[(pw_m - pw_p) *
                                                                                                                                                                                           (_n + 1) + pw_p + 1],
                                                                                                                                                                                 ) if True else pw_pick, pw_pick, pw_p),
                                                                                                                                   [pw_r, 0],
                                                                                                                                   Range(Min(pw_m, pw_c), 1, -1),
                                                                                                                                 ),
                                                                                                                           ), pw_u[3]), pw_u[2]), pw_u[1]),
                    pw_u,
                    pw_i,
                  ), [_n, _n, _r], Range(1, _n, 1)), 3)

rank(_n, _x)

Fold(Function(Apply((pw_r: integer) => Apply(
                                              (pw_m: integer) => Apply((pw_c: integer) => [pw_r +
                                                                                           Fold(
                                                                                                 Function(pw_s +
                                                                                                          (_tables[(pw_m - pw_q) *
                                                                                                                   (_n + 1) + pw_q + 1] if True else 0), pw_s, pw_q),
                                                                                                 0,
                                                                                                 Range(
                                                                                                        pw_x + 1,
                                                                                                        Min(pw_m, pw_c),
                                                                                                        1,
                                                                                                      ),
                                                                                               ), pw_m - pw_x, pw_x], pw_a[3]),
                                              pw_a[2],
                                            ), pw_a[1]), pw_a, pw_x), [0, _n, _n], _x)[1]

valid(_n, _x)

Apply(
       (pw_end: list<integer>) => pw_end[1] == 1 && pw_end[2] == 0 && 1 != 0,
       Fold(Function(
                      Apply((pw_f: integer) => Apply((pw_m: integer) => Apply(
                                                                               (pw_c: integer) => [1, pw_m - pw_y, pw_y] if pw_f == 1 &&
                                                                                                                            pw_y >= 1 &&
                                                                                                                            pw_y <=
                                                                                                                            Min(pw_m, pw_c) &&
                                                                                                                            True else [0, 0, 0],
                                                                               pw_v[3],
                                                                             ), pw_v[2]), pw_v[1]),
                      pw_v,
                      pw_y,
                    ), [1, _n, _n], _x),
     )

tables(_n)

Fold(
      Function(Join(
                     pw_t,
                     Fold(
                           Function(
                                     ReplaceAt(
                                                pw_row,
                                                pw_c + 1,
                                                Fold(
                                                      Function(pw_a +
                                                               (pw_t[(pw_s -
                                                                      pw_p) * (_n +
                                                                           1) +
                                                                     pw_p + 1] if True else 0), pw_a, pw_p),
                                                      0,
                                                      Range(
                                                             1,
                                                             Min(pw_s, pw_c),
                                                             1,
                                                           ),
                                                    ),
                                              ),
                                     pw_row,
                                     pw_c,
                                   ),
                           Map(Function(0, pw_z), Range(0, _n + 1 - 1, 1)),
                           Range(0, _n + 1 - 1, 1),
                         ),
                   ), pw_t, pw_s),
      Fold(
            Function(ReplaceAt(pw_row, pw_c + 1, 1), pw_row, pw_c),
            Map(Function(0, pw_z), Range(0, _n + 1 - 1, 1)),
            Range(0, _n + 1 - 1, 1),
          ),
      Range(1, _n + 1 - 1, 1),
    )

Statistics

On IntegerPartition, called as CombinatorialStat(x, "Name").

AddableCells St000318Cells that can be added to the diagram and leave a partition: one more than the distinct parts.
Aft St001214The size minus the larger of the largest part and the number of parts.
AlladiWeight St000477The last part times the product over consecutive parts of the gap less 1.
ArmOfFirstCellThe arm of cell (1,1): the first part minus one.
BerkovichGarvanRank St001383The sum of a checkerboard filling of the diagram, +1 on cells whose coordinates sum to an even number and -1 on the others.
BorderSkewStandardTableauCount St001387Standard 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 St001603The 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 St001914The partitions that Bulgarian solitaire (remove the first column, insert it as a row) visits from this one, itself included.
BulgarianSolitairePreimageCount St001913The 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 St001912The steps of Bulgarian solitaire before this partition reaches the cycle it ends in.
CartanDeterminant St001571The determinant of the matrix of the minimums of the distinct parts: the least distinct part times the gaps between consecutive distinct parts.
CellsOutsideDurfeeSquare St001382Cells of the diagram outside its Durfee square.
CellsOutsideFirstHook St001175Cells in neither the first row nor the first column: the size minus the hook length of cell (1,1).
CellsOutsideFirstRow St001176Cells of the diagram below the first row: the size minus the largest part.
CellsOutsideLargestStaircase St001384Cells of the diagram outside the largest staircase it contains.
CellsWithEqualArmAndLeg St001924Cells whose arm and leg are equal.
CellsWithZeroLegAndOddArm St000149Cells with leg 0 and an odd arm.
CentralizerSize St000184The 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 St000346The partitions that coarsen this one, whose parts subdivide to give its. Itself included.
CohookLengthProduct St001378The product of the cohook lengths i + j - 1 of the cells.
CompleteHomogeneousSymmetricFunctionAtOne St001562The 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.
ConjugacyClassSize St000182The number of permutations of the same cycle type: n! over the product of the cycle sizes and the factorials of the multiplicities.
ConjugateDistinctParts St000159Distinct part sizes of the conjugate.
ConjugateOddParts St000992Odd parts of the conjugate — equivalently, the distinct part sizes of λ.
ContainingRectangleShortSide St000533The shorter side of the smallest rectangle that contains the diagram: the smaller of the largest part and the number of parts.
ContainingSquareSide St000784The side of the smallest square that contains the diagram: the larger of the largest part and the number of parts.
ContentSum St000509The sum of the contents j - i of the cells.
CornerMajorIndex St000770The 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 St000159Corner cells — parts strictly larger than the next part (the last part always counts).
CouplesAreForeverGrundyValue St000752The 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 St000474The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
CubedStandardTableauCount St000901The cube of the number of standard tableaux.
CubeRootCount St001364The 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.
CyclicSievingPolynomialDegree St001918(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 St000319 St000320The sum of the arms of the cells on the main diagonal.
DiagonalHookLengthProduct St000870The product of the hook lengths of the cells on the main diagonal.
DiagonalInversionDefect St000377Cells whose arm minus leg is neither 0 nor 1.
DiagonalInversions St000378Cells whose arm minus leg is 0 or 1.
DistinctEvenParts St001092Distinct even part sizes.
DistinctEvenSubsetSums St000547Distinct nonzero even sums of sub-multisets of the parts.
DistinctMultiplicities St000897How many different multiplicities the part sizes have.
DistinctOddPartsBelowLargestEvenPart St001588Distinct odd part sizes smaller than the largest even part.
DistinctOddSubsetSums St000549Distinct odd sums of sub-multisets of the parts.
DistinctParts St000159How many distinct part sizes occur.
DistinctPartsEqualToMultiplicity St001940Distinct part sizes equal to their multiplicity.
DistinctRepeatedParts St000257Distinct part sizes that occur at least twice.
DistinctSubsetSums St000548Distinct sums of non-empty sub-multisets of the parts.
DominatedPartitionCount St000321The partitions of the same size that this one dominates (partial sums of the parts at least those of the other), itself included.
DurfeeSquare St000183The side of the Durfee square: the largest d with at least d parts of size at least d.
DysonRank St000145Largest part minus number of parts.
EdgeOrbitCount St001982Orbits 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 St001561The 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 St000997The largest even part minus the number of odd parts.
EvenParts St000142Parts that are even.
ExponensConsonantiae St001128The least common multiple of the parts over their greatest common divisor.
FirstRowHookLengthSum St000867The sum of the hook lengths in the first row: the size plus the binomial of the largest part and 2.
FlooredHalfMultiplicitySum St000150The sum, over part sizes, of half the multiplicity rounded down.
GiniIndex St001541The size plus 1 choose 2, less the sum over the columns of the column length plus 1 choose 2.
GreatestCommonDivisorOfParts St000667The greatest common divisor of the parts.
HalfLargestEvenPart St001587Half the largest even part; 0 if there is none.
HalfPerimeterHalf the perimeter of the bounding rectangle of the Young diagram: largest part plus number of parts.
HalfSumOfEvenParts St001252Half the sum of the even parts.
HookLengthOfFirstCell St000459The hook length of cell (1,1): the largest part plus the number of parts, less one.
HookLengthOfLastDiagonalCell St000460The hook length of the last cell on the main diagonal: that of cell (d,d) for Durfee side d.
HookLengthSquareSum St001943The sum of the squares of all hook lengths.
HookProduct St000179The product of all hook lengths — the denominator in the hook-length formula.
HurwitzNumber St001763Hurwitz'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 St001814The 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 St000514The simple graphs on the points that a permutation of this cycle type maps to themselves: 2 to the number of edge orbits.
InvariantOrientedCycleCount St000510The 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 St000511Subsets 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 St000512Invariant 3-element subsets: three fixed points, a fixed point and a 2-cycle, or a 3-cycle.
InvariantTwoElementSubsetCount St000513Invariant 2-element subsets: a pair of fixed points or a 2-cycle.
IsSelfConjugate St0011221 when λ equals its conjugate, 0 otherwise.
KrewerasNumber St000517The 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 St001967The coefficient of the monomial in the compositional-inverse series: n (n + 1) ... (n + length - 2) over the factorials of the multiplicities.
LargestEvenPart St000995The largest even part; 0 if there is none.
LargestMissingPart St001392The largest integer below the largest part that is not a part; 0 if every smaller integer is.
LargestMultiplicity St001933The largest multiplicity of a part.
LargestPart St000147The largest part.
LargestRepeatedPart St000143The largest part that occurs at least twice; 0 if the parts are distinct.
LargestShiftedPart St000384The largest part after adding i - 1 to the i-th part.
LargestStaircaseSide St000783 St001432The side of the largest staircase (k, k-1, ..., 1) inside the diagram.
LeastCommonMultipleOfParts St000668The least common multiple of the parts.
LegOfFirstCellThe leg of cell (1,1): the number of parts minus one.
LengthCrank St000146The number of parts when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
LogarithmicDerivativeCoefficient St001968The 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 St000480The partitions of the same size that this one covers in dominance order: the upper covers of its conjugate.
MaximalRectangleHalfPerimeter St000380Half the perimeter of the rectangle inside the diagram that has the greatest perimeter.
MaximalRookPlacementCount St001659The ways to place as many non-attacking rooks as possible on the Ferrers board of the diagram.
MinimalSubpartitionSizeDifference St000835The smallest difference in size between the two halves of a split of the parts into two sub-multisets.
MonotoneHurwitzNumber St001934The 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 St000048The multinomial coefficient of the size over the parts: the size factorial over the product of the factorials of the parts.
MultipartitionCount St000933Tuples of partitions whose sizes are the parts: the product of the partition numbers of the parts.
MultiplicityFactorialProduct St000706The product of the factorials of the multiplicities of the parts.
MultiplicityOfLargestPart St000993How many parts equal the largest.
MultiplicityOfSmallestPart St000160How many parts equal the smallest.
NecklaceCount St001605The 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 St001714The distinct sub-multisets of the parts that, read as partitions, do not dominate their own conjugate.
OddParts St000148Parts that are odd.
OddPartsBelowLargestEvenPart St001586Odd parts smaller than the largest even part.
OrderedRefinementCount St000935The partitions whose parts cut, in order, into consecutive blocks summing to the parts of this one.
PairsOfDistinctParts St000175Pairs of parts that differ in size: the degree of the polynomial counting semistandard tableaux of the stretched shape.
PairwiseGreatestCommonDivisorSum St001961The sum of the greatest common divisors of all pairs of parts.
PairwiseProductSum St000567The sum of the products of all pairs of parts.
PartCount St000010The number of parts.
PartFactorialProduct St000707The product of the factorials of the parts.
PartitionSize St000228The sum of the parts.
PartProduct St000708The product of the parts.
PartRange St000225The largest part minus the smallest.
PartRearrangementCount St000278Distinct rearrangements of the parts, the compositions that sort to this partition: the number of parts factorial over the factorials of the multiplicities.
PartsAtLeastTwo St001280Parts of size at least 2.
PartsEqualOne St000475Parts equal to 1.
PartsEqualToMultiplicity St001939Parts equal to their own multiplicity.
PartsEqualToNextPart St001091Parts whose next smaller part has the same size: the number of parts less the distinct parts.
PartsEqualTwo St001657Parts equal to 2.
PartsExceedingOnes St000473Parts larger than the number of parts equal to 1.
PartsNotCongruentToOneModuloThree St001251Parts that are not congruent to 1 modulo 3.
PartsNotCongruentToTwoModuloThree St001247Parts that are not congruent to 2 modulo 3.
PartsNotCongruentToZeroModuloThree St001250Parts that are not divisible by 3.
PartSquareProduct St001129The product of the squares of the parts.
PartSquareSum St001127The sum of the squares of the parts.
PlancherelCount St000284The square of the number of standard tableaux: the permutations whose Robinson-Schensted shape is this partition.
PowerSumSymmetricFunctionAtOne St001563The 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 St000345The partitions that refine this one, whose parts subdivide to give theirs. Itself included.
RemovableHorizontalDominoes St000256Parts from which 2 can be subtracted and leave a partition: rows ending in a removable horizontal domino.
RimHooksRemovedToThreeCore St000697The 3-rim hooks removed to reach the 3-core.
RimHooksRemovedToTwoCore St000698The 2-rim hooks removed to reach the 2-core: the size less the core's, over 2, the core read off the abacus.
RookPlacementCount St000532The rook placements, of any number of rooks, on the Ferrers board of the diagram.
RowCellPairs St000566Pairs of cells in the same row: the sum over parts of the binomial coefficient of the part and 2.
RowRemovalGrundyValue St001055The 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 St001389The partitions with as many parts contained in this one.
SeaweedAlgebraDimension St001262The dimension of the maximal parabolic seaweed algebra: half the sum of the squares of the parts and of the size.
SecondAlladiWeight St000478Half 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 St000712Semistandard tableaux of the shape with entries at most 4.
SemistandardTableauCountWithEntriesUpToLength St000704Semistandard tableaux of the shape with entries at most the number of parts.
SemistandardTableauCountWithEntriesUpToSize St000705Semistandard tableaux of the shape with entries at most the size of the partition.
SemistandardTableauCountWithEntriesUpToThree St000715Semistandard tableaux of the shape with entries at most 3.
SemistandardTableauCountWithEntriesUpToTwo St000714Semistandard tableaux of the shape with entries at most 2: the product of 2 + j - i over the cells, over the hook product.
SetPartitionsWithBlockSizesCount St000049The 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 St001593Standard 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 St000531The 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 St001484Parts that occur exactly once.
SingletonRefinementCount St000913The chains of single splits (a part into two) from this partition down to all ones.
SmallestMissingPart St000759The smallest positive integer that is not a part.
SmallestNonRepeatedInteger St001568The smallest positive integer that is not a part at least twice.
SmirnovWordCount St001097The 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 St000003The number of standard Young tableaux of the shape: the size factorial over the hook product.
StandardTableauCountWithMajorIndexCongruentToOneModuloSize St001440The standard Young tableaux whose major index leaves remainder 1 on division by the size of the shape (never, when the size is 1).
StarTranspositionFactorizationCount St001936Minimal transitive factorizations of a permutation of this cycle type into star transpositions: the product of the parts times (n + 1) ... (n + length - 2).
SubpartitionCount St000108The partitions contained in this one.
SubsetGreatestCommonDivisorSum St001985The 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 St001248The sum of the even parts.
SumOfHookLengths St000869The total of all hook lengths.
SumOfOddParts St001249The sum of the odd parts.
SumOfPartsAtLeastTwo St001279The sum of the parts that are at least 2.
SymmetricDiagonalHooks St001525Cells on the main diagonal whose arm and leg are equal.
SymplecticDimensionRankThree St000716The dimension of the irreducible representation of Sp(6) with this highest weight; 0 if the partition has more than 3 parts.
SymplecticDimensionRankTwo St000713The dimension of the irreducible representation of Sp(4) with this highest weight; 0 if the partition has more than 2 parts.
TransitiveMonotoneFactorizationCount St001938Transitive monotone factorizations of genus zero of a permutation of this cycle type, into n + length - 2 transpositions.
TwelveTimesMajorIndexVarianceOfStandardTableaux St001178Twelve 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 St001177Twice 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 St000481The partitions of the same size that cover this one in dominance order.
WeightedSize St000185The sum of the parts weighted by their row, counting rows from 0.
YoungLatticeCoveringRelationCount St001360The covering relations of Young's lattice among the partitions contained in this one: the sum of their corners.

Maps

Into and out of IntegerPartition, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

From IntegerPartition

BinaryWordIntegerPartition → BinaryWordInjectionMp00095The shape's boundary read from the first row down: 1 for a step down, 0 for a step left.
BulgarianSolitaireIntegerPartition → IntegerPartitionGraded mapMp00308A move of Bulgarian solitaire: the first column of the Ferrers diagram becomes a new row.
GlaisherFranklinIntegerPartition → IntegerPartitionbijectioninverseBijectionGraded mapMp00312The Glaisher-Franklin bijection: even part sizes halved become repeated parts, and repeated odd parts merge into their binary multiples.
InitialTableauIntegerPartition → StandardTableauInjectionGraded mapMp00042The standard tableau of the shape filled row by row with 1, 2, ..., n.
InverseGlaisherFranklinIntegerPartition → IntegerPartitionbijectioninverseBijectionGraded mapMp00313The inverse of GlaisherFranklin: repeated parts merge into doubled parts, and a leftover part splits into copies of its odd part.
MinimalDyckPathIntegerPartition → DyckPathInjectionMp00043The shortest Dyck path tracing the shape of the Ferrers diagram.
OddPartsWordIntegerPartition → BinaryWordSurjectionMp00317The binary word marking which parts are odd, the parts taken largest first.
ParallelogramPolyominoDyckPathIntegerPartition → DyckPathInjectionMp00230The Dyck path of the partition read as a parallelogram polyomino, one column per part.
ReadingTableauIntegerPartition → StandardTableauInjectionGraded mapMp00045The recording tableau of the row reading word of the shape filled row by row.
RemoveFirstPartIntegerPartition → IntegerPartitionSurjectionMp00202The partition without its largest part: its first row removed.
TransposePartitionIntegerPartition → IntegerPartitioninvolutionBijectionGraded mapInvolutionMp00044The conjugate partition: column i of the Ferrers diagram becomes row i.

To IntegerPartition

BlockSizePartitionSetPartition → IntegerPartitionsurjectionSurjectionGraded mapMp00079The partition of n a set partition's block sizes form, largest first.
ConjugateAfterCycleTypePermutation → IntegerPartitionThe conjugate partition of the cycle type.
CutOutPartitionDyckPath → IntegerPartitionInjectionMp00027The partition cut out of the staircase by the path: part i is the number of downs before the (n + 1 - i)-th up step.
CycleTypePermutation → IntegerPartitionsurjectionSurjectionGraded mapMp00108The multiset of cycle lengths, as a partition.
NegativeCycleTypeSignedPermutation → IntegerPartitionSurjectionMp00169The lengths of the negative cycles, largest first: those of the underlying permutation with an odd number of negative entries.
PositiveCycleTypeSignedPermutation → IntegerPartitionSurjectionMp00166The lengths of the positive cycles, largest first: those of the underlying permutation with an even number of negative entries.
RskShapePermutation → IntegerPartitionsurjectionSurjectionGraded mapMp00060The common shape of the RSK tableaux: the row lengths, as a partition of n.
SortedPartitionIntegerComposition → IntegerPartitionsurjectionSurjectionGraded mapMp00040The partition a composition's parts form, sorted largest first.
TableauShapeStandardTableau → IntegerPartitionsurjectionSurjectionGraded mapMp00083The shape of a standard tableau: its row lengths.

21 more in FindStat, not implemented here

Mp00021CorePartition → IntegerPartitionGraded mapBijectionto bounded partition
Mp00022CorePartition → IntegerPartitionSurjectionto partition
Mp00037LabeledGraph → IntegerPartitionSurjectionGraded mapto partition of connected components
Mp00077SemistandardTableau → IntegerPartitionSurjectionGraded mapshape
Mp00110Posets → IntegerPartitionSurjectionGraded mapGreene-Kleitman invariant
Mp00179IntegerPartition → SkewPartitionInjectionGraded mapto skew partition
Mp00182SkewPartition → IntegerPartitionSurjectionouter shape
Mp00183SkewPartition → IntegerPartitionSurjectioninner shape
Mp00186SkewPartition → IntegerPartitionSurjectionGraded mapdominating partition
Mp00204Permutation → IntegerPartitionSurjectionGraded mapLLPS
Mp00225SemistandardTableau → IntegerPartitionSurjectionweight
Mp00251LabeledGraph → IntegerPartitionclique sizes
Mp00275LabeledGraph → IntegerPartitionSurjectionto edge-partition of connected components
Mp00276LabeledGraph → IntegerPartitionto edge-partition of biconnected components
Mp00306Posets → IntegerPartitionrowmotion cycle type
Mp00307Posets → IntegerPartitionpromotion cycle type
Mp00311PlanePartition → IntegerPartitionGraded mapto partition
Mp00321IntegerPartition → IntegerPartitionBijectionGraded mapInvolution2-conjugate
Mp00322IntegerPartition → IntegerPartitionBijectionGraded mapLoehr-Warrington
Mp00323IntegerPartition → IntegerPartitionBijectionGraded mapLoehr-Warrington inverse
Mp00332Posets → IntegerPartitionSurjectionGraded mapJordan block partition

Enumeration

Examples

Basic #

The partitions of 8 into at most 3 parts

live ↗

\mathrm{IntegerPartitions}(8, 3)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([8]), {\operatorname{IntegerPartition}}([7, 1]), {\operatorname{IntegerPartition}}([6, 2]), {\operatorname{IntegerPartition}}([6, 1, 1]), {\operatorname{IntegerPartition}}([5, 3]), {\operatorname{IntegerPartition}}([5, 2, 1]), {\operatorname{IntegerPartition}}([4, 4]), {\operatorname{IntegerPartition}}([4, 3, 1]), {\operatorname{IntegerPartition}}([4, 2, 2]), {\operatorname{IntegerPartition}}([3, 3, 2])\bigr\rbrack

Partitions of 5 come out largest part first, ending at all ones

live ↗

\mathrm{IntegerPartitions}(5)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([5]), {\operatorname{IntegerPartition}}([4, 1]), {\operatorname{IntegerPartition}}([3, 2]), {\operatorname{IntegerPartition}}([3, 1, 1]), {\operatorname{IntegerPartition}}([2, 2, 1]), {\operatorname{IntegerPartition}}([2, 1, 1, 1]), {\operatorname{IntegerPartition}}([1, 1, 1, 1, 1])\bigr\rbrack

$p(6)$

live ↗

\mathrm{Count}(\mathrm{IntegerPartitions}(6))

Out

11

the single part $6$

live ↗

\mathrm{IntegerPartitions}(6)[1]

Out

{\operatorname{IntegerPartition}}([6])

Scope #

Exactly 3 parts

live ↗

\mathrm{IntegerPartitions}(8, \bigl\lbrack3\bigr\rbrack)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([6, 1, 1]), {\operatorname{IntegerPartition}}([5, 2, 1]), {\operatorname{IntegerPartition}}([4, 3, 1]), {\operatorname{IntegerPartition}}([4, 2, 2]), {\operatorname{IntegerPartition}}([3, 3, 2])\bigr\rbrack

Parts restricted to $\{1, 2, 5\}$

live ↗

\mathrm{IntegerPartitions}(8, \mathrm{All}, \bigl\lbrack1, 2, 5\bigr\rbrack)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([5, 2, 1]), {\operatorname{IntegerPartition}}([5, 1, 1, 1]), {\operatorname{IntegerPartition}}([2, 2, 2, 2]), {\operatorname{IntegerPartition}}([2, 2, 2, 1, 1]), {\operatorname{IntegerPartition}}([2, 2, 1, 1, 1, 1]), {\operatorname{IntegerPartition}}([2, 1, 1, 1, 1, 1, 1]), {\operatorname{IntegerPartition}}([1, 1, 1, 1, 1, 1, 1, 1])\bigr\rbrack

live ↗

\mathrm{IntegerPartitions}(6, \bigl\lbrack2, ∞, 2\bigr\rbrack)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([5, 1]), {\operatorname{IntegerPartition}}([4, 2]), {\operatorname{IntegerPartition}}([3, 3]), {\operatorname{IntegerPartition}}([3, 1, 1, 1]), {\operatorname{IntegerPartition}}([2, 2, 1, 1]), {\operatorname{IntegerPartition}}([1, 1, 1, 1, 1, 1])\bigr\rbrack

live ↗

\mathrm{IntegerPartitions}(3, 10, \bigl\lbrack1, \frac{1}{3}, \frac{3}{4}\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\frac{3}{4}, \frac{3}{4}, \frac{3}{4}, \frac{3}{4}\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\bigr\rbrack

live ↗

\mathrm{IntegerPartitions}(5, 10, \bigl\lbrack1, -1\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack-1, -1, 1, 1, 1, 1, 1, 1, 1\bigr\rbrack, \bigl\lbrack-1, 1, 1, 1, 1, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 1, 1, 1\bigr\rbrack\bigr\rbrack

Applications #

The ways to make change for a dollar from 1, 5, 10, 25 and 50 cent coins

live ↗

\mathrm{Count}(\mathrm{IntegerPartitions}(100, \mathrm{All}, \bigl\lbrack1, 5, 10, 25, 50\bigr\rbrack))

Out

292

live ↗

\mathrm{IntegerPartitions}(156, 10, \bigl\lbrack1, 5, 10, 25\bigr\rbrack)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([25, 25, 25, 25, 25, 25, 5, 1]), {\operatorname{IntegerPartition}}([25, 25, 25, 25, 25, 10, 10, 10, 1]), {\operatorname{IntegerPartition}}([25, 25, 25, 25, 25, 10, 10, 5, 5, 1])\bigr\rbrack

Properties #

The family has $p(8) = 22$ members, compute-engine's NPartition

live ↗

\mathrm{NPartition}(8)

Out

22

The five partitions of 4, in reverse-lexicographic order

live ↗

\mathrm{IntegerPartitions}(4)

Out

\bigl\lbrack{\operatorname{IntegerPartition}}([4]), {\operatorname{IntegerPartition}}([3, 1]), {\operatorname{IntegerPartition}}([2, 2]), {\operatorname{IntegerPartition}}([2, 1, 1]), {\operatorname{IntegerPartition}}([1, 1, 1, 1])\bigr\rbrack

There are $p(10) = 42$ partitions of 10

live ↗

\mathrm{Length}(\mathrm{IntegerPartitions}(10))

Out

42

live ↗

\mathrm{Count}(\mathrm{IntegerPartitions}(10))

Out

42

Neat examples #

Euler: partitions into odd parts are as many as partitions into distinct parts, $q(10) = 10$ . See DistinctPartitions

live ↗

\mathrm{Count}(\mathrm{IntegerPartitions}(10, \mathrm{All}, \bigl\lbrack1, 3, 5, 7, 9\bigr\rbrack))

Out

10

Statistics #

partitions of 7 by largest part

live ↗

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{IntegerPartitions}(7), \text{LargestPart}))

Out

\operatorname{Tuple}(\bigl\lbrack7, 6, 5, 4, 3, 2, 1\bigr\rbrack,\bigl\lbrack1, 1, 2, 3, 4, 3, 1\bigr\rbrack)

partitions of 7 by how many distinct part sizes they use

live ↗

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{IntegerPartitions}(7), \text{DistinctParts}))

Out

\operatorname{Tuple}(\bigl\lbrack1, 2, 3\bigr\rbrack,\bigl\lbrack2, 11, 2\bigr\rbrack)

About

Examples	18
Checked against	Wolfram Language 15 agreeCombinatorica 9 agreemathlib4 5 agree1 error
Crosswalk	Wikipedia ×2 MathWorld Wikidata Britannica OEIS ×6 Wolfram Language ×2Combinatorica ×2SageMath SymPy mathlib4

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica's Partitions[n, k] bounds the LARGEST part by k; ours (and Wolfram's) bound the number of parts, so the template transposes each partition and restores descending lexicographic order. A list {k}, {kmin, kmax} or {kmin, kmax, dk} selects the partitions of Partitions[n] by their number of parts.

mappedmathlib4packages/oracle/src/mappings.ts

Nat.Partition n enumerated (Fintype), each partition's parts sorted largest first and the lot in reverse lexicographic order, as ours lists them; the one-argument form only

IntegerPartitions ​

Definition

Statistics

Maps

From IntegerPartition

To IntegerPartition

Enumeration

Examples

Implementation

IntegerPartitions