PartitionsIntoKParts
The integer partitions of into exactly positive parts.
✓ checked against that system · ! checked, with disagreements (hover a chip)OEIS
A072233✓PartitionsIntoKParts(n, k)partitions of with exactly partsTyped signatures
| Type | Package | Applies |
|---|---|---|
(integer<0..>, integer<0..>) -> indexed_collection<integer_partition> | combinatorics | always |
Definition
Epsil, as the implementation runs it.
count(_n, _k)
_tables[_n * (Min(_k, _n + 1) + 1) * (_n + 1) +
Min(_k, _n + 1) * (_n + 1) + _n + 1]unrank(_n, _k, _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_c -
pw_c %
(_n + 1)) /
(_n + 1) -
1) *
(_n + 1) +
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) *
(Min(_k, _n + 1) +
1) *
(_n + 1) +
((pw_c -
pw_c %
(_n + 1)) /
(_n + 1) -
1) *
(_n + 1) +
pw_p +
1],
) if (pw_c -
pw_c %
(_n + 1)) /
(_n + 1) >=
1 else pw_pick, pw_pick, pw_p),
[pw_r, 0],
Range(Min(pw_m, pw_c %
(_n + 1)), 1, -1),
),
), pw_u[3]), pw_u[2]), pw_u[1]), pw_u, pw_i),
[_n, Min(_k, _n + 1) * (_n + 1) + _n, _r],
Range(1, _n, 1),
), 3)rank(_n, _k, _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) *
(Min(_k, _n + 1) +
1) *
(_n + 1) +
((pw_c -
pw_c %
(_n + 1)) /
(_n + 1) -
1) *
(_n + 1) +
pw_q +
1] if (pw_c -
pw_c %
(_n + 1)) /
(_n + 1) >=
1 else 0), pw_s, pw_q),
0,
Range(
pw_x + 1,
Min(pw_m, pw_c %
(_n + 1)),
1,
),
),
pw_m - pw_x,
((pw_c -
pw_c %
(_n + 1)) /
(_n + 1) -
1) *
(_n + 1) +
pw_x,
], pw_a[3]), pw_a[2]), pw_a[1]), pw_a, pw_x),
[0, _n, Min(_k, _n + 1) * (_n + 1) + _n],
_x,
)[1]valid(_n, _k, _x)
Apply(
(pw_end: list<integer>) => pw_end[1] == 1 && pw_end[2] == 0 && (1 if (pw_end[3] -
pw_end[3] %
(_n +
1)) /
(_n +
1) ==
0 else 0) !=
0,
Fold(Function(Apply(
(pw_f: integer) => Apply((pw_m: integer) => Apply((pw_c: integer) => [
1,
pw_m - pw_y,
((pw_c -
pw_c %
(_n + 1)) /
(_n + 1) -
1) *
(_n + 1) +
pw_y,
] if pw_f == 1 &&
pw_y >= 1 &&
pw_y <=
Min(pw_m, pw_c %
(_n + 1)) &&
(pw_c -
pw_c %
(_n + 1)) /
(_n + 1) >=
1 else [0, 0, 0], pw_v[3]), pw_v[2]),
pw_v[1],
), pw_v, pw_y), [
1,
_n,
Min(_k, _n + 1) * (_n + 1) + _n,
], _x),
)tables(_n, _k)
Fold(
Function(Join(
pw_t,
Fold(
Function(ReplaceAt(
pw_row,
pw_c + 1,
Fold(
Function(pw_a +
(pw_t[(pw_s - pw_p) *
(Min(_k, _n +
1) +
1) *
(_n + 1) +
((pw_c - pw_c %
(_n +
1)) /
(_n + 1) - 1) * (_n +
1) +
pw_p + 1] if (pw_c -
pw_c %
(_n +
1)) / (_n +
1) >=
1 else 0), pw_a, pw_p),
0,
Range(1, Min(pw_s, pw_c %
(_n + 1)), 1),
),
), pw_row, pw_c),
Map(
Function(0, pw_z),
Range(
0,
(Min(_k, _n + 1) + 1) * (_n + 1) - 1,
1,
),
),
Range(0, (Min(_k, _n + 1) + 1) * (_n + 1) - 1, 1),
),
), pw_t, pw_s),
Fold(
Function(
ReplaceAt(pw_row, pw_c + 1, 1 if (pw_c - pw_c % (_n + 1)) /
(_n + 1) == 0 else 0),
pw_row,
pw_c,
),
Map(
Function(0, pw_z),
Range(0, (Min(_k, _n + 1) + 1) * (_n + 1) - 1, 1),
),
Range(0, (Min(_k, _n + 1) + 1) * (_n + 1) - 1, 1),
),
Range(1, _n + 1 - 1, 1),
)Statistics
On IntegerPartition, called as CombinatorialStat(x, "Name").
- AddableCellsSt000318Cells that can be added to the diagram and leave a partition: one more than the distinct parts.
- AftSt001214The size minus the larger of the largest part and the number of parts.
- AlladiWeightSt000477The last part times the product over consecutive parts of the gap less 1.
- ArmOfFirstCellThe arm of cell (1,1): the first part minus one.
- BerkovichGarvanRankSt001383The sum of a checkerboard filling of the diagram, +1 on cells whose coordinates sum to an even number and -1 on the others.
- BorderSkewStandardTableauCountSt001387Standard 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).
- BraceletCountSt001603The 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.
- BulgarianSolitaireOrbitSizeSt001914The partitions that Bulgarian solitaire (remove the first column, insert it as a row) visits from this one, itself included.
- BulgarianSolitairePreimageCountSt001913The 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.
- BulgarianSolitairePreperiodSt001912The steps of Bulgarian solitaire before this partition reaches the cycle it ends in.
- CartanDeterminantSt001571The determinant of the matrix of the minimums of the distinct parts: the least distinct part times the gaps between consecutive distinct parts.
- CellsOutsideDurfeeSquareSt001382Cells of the diagram outside its Durfee square.
- CellsOutsideFirstHookSt001175Cells in neither the first row nor the first column: the size minus the hook length of cell (1,1).
- CellsOutsideFirstRowSt001176Cells of the diagram below the first row: the size minus the largest part.
- CellsOutsideLargestStaircaseSt001384Cells of the diagram outside the largest staircase it contains.
- CellsWithEqualArmAndLegSt001924Cells whose arm and leg are equal.
- CellsWithZeroLegAndOddArmSt000149Cells with leg 0 and an odd arm.
- CentralizerSizeSt000184The 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.
- CoarseningCountSt000346The partitions that coarsen this one, whose parts subdivide to give its. Itself included.
- CohookLengthProductSt001378The product of the cohook lengths i + j - 1 of the cells.
- CompleteHomogeneousSymmetricFunctionAtOneSt001562The 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.
- ConjugacyClassSizeSt000182The number of permutations of the same cycle type: n! over the product of the cycle sizes and the factorials of the multiplicities.
- ConjugateDistinctPartsSt000159Distinct part sizes of the conjugate.
- ConjugateOddPartsSt000992Odd parts of the conjugate — equivalently, the distinct part sizes of λ.
- ContainingRectangleShortSideSt000533The shorter side of the smallest rectangle that contains the diagram: the smaller of the largest part and the number of parts.
- ContainingSquareSideSt000784The side of the smallest square that contains the diagram: the larger of the largest part and the number of parts.
- ContentSumSt000509The sum of the contents j - i of the cells.
- CornerMajorIndexSt000770The sum, over the corners, of the step number along the boundary path from the bottom left: (number of parts - i) + the i-th part.
- CornersSt000159Corner cells — parts strictly larger than the next part (the last part always counts).
- CouplesAreForeverGrundyValueSt000752The 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.
- CrankSt000474The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
- CubedStandardTableauCountSt000901The cube of the number of standard tableaux.
- CubeRootCountSt001364The 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.
- CyclicSievingPolynomialDegreeSt001918(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.
- DiagonalArmSumSt000319St000320The sum of the arms of the cells on the main diagonal.
- DiagonalHookLengthProductSt000870The product of the hook lengths of the cells on the main diagonal.
- DiagonalInversionDefectSt000377Cells whose arm minus leg is neither 0 nor 1.
- DiagonalInversionsSt000378Cells whose arm minus leg is 0 or 1.
- DistinctEvenPartsSt001092Distinct even part sizes.
- DistinctEvenSubsetSumsSt000547Distinct nonzero even sums of sub-multisets of the parts.
- DistinctMultiplicitiesSt000897How many different multiplicities the part sizes have.
- DistinctOddPartsBelowLargestEvenPartSt001588Distinct odd part sizes smaller than the largest even part.
- DistinctOddSubsetSumsSt000549Distinct odd sums of sub-multisets of the parts.
- DistinctPartsSt000159How many distinct part sizes occur.
- DistinctPartsEqualToMultiplicitySt001940Distinct part sizes equal to their multiplicity.
- DistinctRepeatedPartsSt000257Distinct part sizes that occur at least twice.
- DistinctSubsetSumsSt000548Distinct sums of non-empty sub-multisets of the parts.
- DominatedPartitionCountSt000321The partitions of the same size that this one dominates (partial sums of the parts at least those of the other), itself included.
- DurfeeSquareSt000183The side of the Durfee square: the largest d with at least d parts of size at least d.
- DysonRankSt000145Largest part minus number of parts.
- EdgeOrbitCountSt001982Orbits 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.
- ElementarySymmetricFunctionAtOneSt001561The 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.
- EvenOddCrankSt000997The largest even part minus the number of odd parts.
- EvenPartsSt000142Parts that are even.
- ExponensConsonantiaeSt001128The least common multiple of the parts over their greatest common divisor.
- FirstRowHookLengthSumSt000867The sum of the hook lengths in the first row: the size plus the binomial of the largest part and 2.
- FlooredHalfMultiplicitySumSt000150The sum, over part sizes, of half the multiplicity rounded down.
- GiniIndexSt001541The size plus 1 choose 2, less the sum over the columns of the column length plus 1 choose 2.
- GreatestCommonDivisorOfPartsSt000667The greatest common divisor of the parts.
- HalfLargestEvenPartSt001587Half 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.
- HalfSumOfEvenPartsSt001252Half the sum of the even parts.
- HookLengthOfFirstCellSt000459The hook length of cell (1,1): the largest part plus the number of parts, less one.
- HookLengthOfLastDiagonalCellSt000460The hook length of the last cell on the main diagonal: that of cell (d,d) for Durfee side d.
- HookLengthSquareSumSt001943The sum of the squares of all hook lengths.
- HookProductSt000179The product of all hook lengths — the denominator in the hook-length formula.
- HurwitzNumberSt001763Hurwitz'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.
- InterlacingPartitionCountSt001814The 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.
- InvariantGraphCountSt000514The simple graphs on the points that a permutation of this cycle type maps to themselves: 2 to the number of edge orbits.
- InvariantOrientedCycleCountSt000510The 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.
- InvariantSubsetCountSt000511Subsets of the points left invariant by a permutation with these cycle lengths: each cycle is in or out, 2 to the number of cycles.
- InvariantThreeElementSubsetCountSt000512Invariant 3-element subsets: three fixed points, a fixed point and a 2-cycle, or a 3-cycle.
- InvariantTwoElementSubsetCountSt000513Invariant 2-element subsets: a pair of fixed points or a 2-cycle.
- IsSelfConjugateSt0011221 when λ equals its conjugate, 0 otherwise.
- KrewerasNumberSt000517The 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.
- LagrangeInversionCoefficientSt001967The coefficient of the monomial in the compositional-inverse series: n (n + 1) ... (n + length - 2) over the factorials of the multiplicities.
- LargestEvenPartSt000995The largest even part; 0 if there is none.
- LargestMissingPartSt001392The largest integer below the largest part that is not a part; 0 if every smaller integer is.
- LargestMultiplicitySt001933The largest multiplicity of a part.
- LargestPartSt000147The largest part.
- LargestRepeatedPartSt000143The largest part that occurs at least twice; 0 if the parts are distinct.
- LargestShiftedPartSt000384The largest part after adding i - 1 to the i-th part.
- LargestStaircaseSideSt000783St001432The side of the largest staircase (k, k-1, ..., 1) inside the diagram.
- LeastCommonMultipleOfPartsSt000668The least common multiple of the parts.
- LegOfFirstCellThe leg of cell (1,1): the number of parts minus one.
- LengthCrankSt000146The number of parts when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).
- LogarithmicDerivativeCoefficientSt001968The 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.
- LowerCoverCountInDominanceOrderSt000480The partitions of the same size that this one covers in dominance order: the upper covers of its conjugate.
- MaximalRectangleHalfPerimeterSt000380Half the perimeter of the rectangle inside the diagram that has the greatest perimeter.
- MaximalRookPlacementCountSt001659The ways to place as many non-attacking rooks as possible on the Ferrers board of the diagram.
- MinimalSubpartitionSizeDifferenceSt000835The smallest difference in size between the two halves of a split of the parts into two sub-multisets.
- MonotoneHurwitzNumberSt001934The 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.
- MultinomialOfPartsSt000048The multinomial coefficient of the size over the parts: the size factorial over the product of the factorials of the parts.
- MultipartitionCountSt000933Tuples of partitions whose sizes are the parts: the product of the partition numbers of the parts.
- MultiplicityFactorialProductSt000706The product of the factorials of the multiplicities of the parts.
- MultiplicityOfLargestPartSt000993How many parts equal the largest.
- MultiplicityOfSmallestPartSt000160How many parts equal the smallest.
- NecklaceCountSt001605The 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.
- NonDominatingSubmultisetCountSt001714The distinct sub-multisets of the parts that, read as partitions, do not dominate their own conjugate.
- OddPartsSt000148Parts that are odd.
- OddPartsBelowLargestEvenPartSt001586Odd parts smaller than the largest even part.
- OrderedRefinementCountSt000935The partitions whose parts cut, in order, into consecutive blocks summing to the parts of this one.
- PairsOfDistinctPartsSt000175Pairs of parts that differ in size: the degree of the polynomial counting semistandard tableaux of the stretched shape.
- PairwiseGreatestCommonDivisorSumSt001961The sum of the greatest common divisors of all pairs of parts.
- PairwiseProductSumSt000567The sum of the products of all pairs of parts.
- PartCountSt000010The number of parts.
- PartFactorialProductSt000707The product of the factorials of the parts.
- PartitionSizeSt000228The sum of the parts.
- PartProductSt000708The product of the parts.
- PartRangeSt000225The largest part minus the smallest.
- PartRearrangementCountSt000278Distinct rearrangements of the parts, the compositions that sort to this partition: the number of parts factorial over the factorials of the multiplicities.
- PartsAtLeastTwoSt001280Parts of size at least 2.
- PartsEqualOneSt000475Parts equal to 1.
- PartsEqualToMultiplicitySt001939Parts equal to their own multiplicity.
- PartsEqualToNextPartSt001091Parts whose next smaller part has the same size: the number of parts less the distinct parts.
- PartsEqualTwoSt001657Parts equal to 2.
- PartsExceedingOnesSt000473Parts larger than the number of parts equal to 1.
- PartsNotCongruentToOneModuloThreeSt001251Parts that are not congruent to 1 modulo 3.
- PartsNotCongruentToTwoModuloThreeSt001247Parts that are not congruent to 2 modulo 3.
- PartsNotCongruentToZeroModuloThreeSt001250Parts that are not divisible by 3.
- PartSquareProductSt001129The product of the squares of the parts.
- PartSquareSumSt001127The sum of the squares of the parts.
- PlancherelCountSt000284The square of the number of standard tableaux: the permutations whose Robinson-Schensted shape is this partition.
- PowerSumSymmetricFunctionAtOneSt001563The 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.
- RefinementCountSt000345The partitions that refine this one, whose parts subdivide to give theirs. Itself included.
- RemovableHorizontalDominoesSt000256Parts from which 2 can be subtracted and leave a partition: rows ending in a removable horizontal domino.
- RimHooksRemovedToThreeCoreSt000697The 3-rim hooks removed to reach the 3-core.
- RimHooksRemovedToTwoCoreSt000698The 2-rim hooks removed to reach the 2-core: the size less the core's, over 2, the core read off the abacus.
- RookPlacementCountSt000532The rook placements, of any number of rooks, on the Ferrers board of the diagram.
- RowCellPairsSt000566Pairs of cells in the same row: the sum over parts of the binomial coefficient of the part and 2.
- RowRemovalGrundyValueSt001055The 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.
- SameLengthSubpartitionCountSt001389The partitions with as many parts contained in this one.
- SeaweedAlgebraDimensionSt001262The dimension of the maximal parabolic seaweed algebra: half the sum of the squares of the parts and of the size.
- SecondAlladiWeightSt000478Half 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.
- SemistandardTableauCountWithEntriesUpToFourSt000712Semistandard tableaux of the shape with entries at most 4.
- SemistandardTableauCountWithEntriesUpToLengthSt000704Semistandard tableaux of the shape with entries at most the number of parts.
- SemistandardTableauCountWithEntriesUpToSizeSt000705Semistandard tableaux of the shape with entries at most the size of the partition.
- SemistandardTableauCountWithEntriesUpToThreeSt000715Semistandard tableaux of the shape with entries at most 3.
- SemistandardTableauCountWithEntriesUpToTwoSt000714Semistandard tableaux of the shape with entries at most 2: the product of 2 + j - i over the cells, over the hook product.
- SetPartitionsWithBlockSizesCountSt000049The 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.
- ShiftedStandardTableauCountSt001593Standard 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.
- ShorterSideRookPlacementCountSt000531The 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.
- SingletonPartsSt001484Parts that occur exactly once.
- SingletonRefinementCountSt000913The chains of single splits (a part into two) from this partition down to all ones.
- SmallestMissingPartSt000759The smallest positive integer that is not a part.
- SmallestNonRepeatedIntegerSt001568The smallest positive integer that is not a part at least twice.
- SmirnovWordCountSt001097The 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.
- StandardTableauCountSt000003The number of standard Young tableaux of the shape: the size factorial over the hook product.
- StandardTableauCountWithMajorIndexCongruentToOneModuloSizeSt001440The standard Young tableaux whose major index leaves remainder 1 on division by the size of the shape (never, when the size is 1).
- StarTranspositionFactorizationCountSt001936Minimal transitive factorizations of a permutation of this cycle type into star transpositions: the product of the parts times (n + 1) ... (n + length - 2).
- SubpartitionCountSt000108The partitions contained in this one.
- SubsetGreatestCommonDivisorSumSt001985The 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.
- SumOfEvenPartsSt001248The sum of the even parts.
- SumOfHookLengthsSt000869The total of all hook lengths.
- SumOfOddPartsSt001249The sum of the odd parts.
- SumOfPartsAtLeastTwoSt001279The sum of the parts that are at least 2.
- SymmetricDiagonalHooksSt001525Cells on the main diagonal whose arm and leg are equal.
- SymplecticDimensionRankThreeSt000716The dimension of the irreducible representation of Sp(6) with this highest weight; 0 if the partition has more than 3 parts.
- SymplecticDimensionRankTwoSt000713The dimension of the irreducible representation of Sp(4) with this highest weight; 0 if the partition has more than 2 parts.
- TransitiveMonotoneFactorizationCountSt001938Transitive monotone factorizations of genus zero of a permutation of this cycle type, into n + length - 2 transpositions.
- TwelveTimesMajorIndexVarianceOfStandardTableauxSt001178Twelve 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.
- TwiceMeanMajorIndexOfStandardTableauxSt001177Twice 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.
- UpperCoverCountInDominanceOrderSt000481The partitions of the same size that cover this one in dominance order.
- WeightedSizeSt000185The sum of the parts weighted by their row, counting rows from 0.
- YoungLatticeCoveringRelationCountSt001360The 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
Examples
About
| Examples | 2 |
|---|---|
| Checked against | mathlib4 2 agree |
| Crosswalk | OEIS |
Implementation
mapped
mathlib4packages/oracle/src/mappings.tsNat.Partition n enumerated, those with exactly k parts, ordered as IntegerPartitions
See also: IntegerPartitions, PartitionsInBox