PartitionsInBox

The integer partitions fitting in an $a \times b$ box: at most $a$ parts, each at most $b$. Count $\binom{a+b}{a}$.

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PartitionsInBox(a, b)partitions with at most $a$ parts, each at most $b$

Typed signatures

Type	Package	Applies
(integer<0..>, integer<0..>) -> indexed_collection<integer_partition>	combinatorics	always

Domain: Collections

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.
• ArmOfFirstCellThe arm of cell (1,1): the first part minus one.
• CellsOutsideDurfeeSquareSt001382Cells of the diagram outside its Durfee square.
• CellsOutsideLargestStaircaseSt001384Cells of the diagram outside the largest staircase it contains.
• CellsWithEqualArmAndLegSt001924Cells whose arm and leg are equal.
• CohookLengthProductSt001378The product of the cohook lengths i + j - 1 of the cells.
• ConjugateDistinctPartsSt000159Distinct part sizes of the conjugate.
• ConjugateOddPartsSt000992Odd parts of the conjugate — equivalently, the distinct part sizes of λ.
• ContainingSquareSideSt000784The side of the smallest square that contains the diagram: the larger of the largest part and the number of parts.
• CornersSt000159Corner cells — parts strictly larger than the next part (the last part always counts).
• 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).
• DiagonalHookLengthProductSt000870The product of the hook lengths of the cells on the main diagonal.
• DiagonalIndexSt000509The sum of the contents j - i of the cells.
• DiagonalInversionDefectSt000377Cells whose arm minus leg is neither 0 nor 1.
• DiagonalInversionsSt000378Cells whose arm minus leg is 0 or 1.
• DistinctMultiplicitiesSt000897How many different multiplicities the part sizes have.
• DistinctPartsSt000159How many distinct part sizes occur.
• DistinctPartsEqualToMultiplicitySt001940Distinct part sizes equal to their multiplicity.
• DistinctRepeatedPartsSt000257Distinct part sizes that occur at least twice.
• 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.
• EvenPartsSt000142Parts that are even.
• 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.
• 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.
• IsSelfConjugateSt0011221 when λ equals its conjugate, 0 otherwise.
• LargestMultiplicitySt001933The largest multiplicity of a part.
• LargestPartSt000147The largest part.
• LargestRepeatedPartSt000143The largest part that occurs at least twice; 0 if the parts are distinct.
• LargestStaircaseSideSt000783St001432The side of the largest staircase (k, k-1, ..., 1) inside the diagram.
• LegOfFirstCellThe leg of cell (1,1): the number of parts minus one.
• MultiplicityOfLargestPartSt000993How many parts equal the largest.
• MultiplicityOfSmallestPartSt000160How many parts equal the smallest.
• OddPartsSt000148Parts that are odd.
• PairwiseProductSumSt000567The sum of the products of all pairs of parts.
• PartsAtLeastTwoSt001280Parts of size at least 2.
• PartsEqualOneSt000475Parts equal to 1.
• PartsEqualToMultiplicitySt001939Parts equal to their own multiplicity.
• PartsExceedingOnesSt000473Parts larger than the number of parts equal to 1.
• PerimeterThe perimeter of the Young diagram: largest part plus number of parts.
• SingletonPartsSt001484Parts that occur exactly once.
• SmallestMissingPartSt000759The smallest positive integer that is not a part.
• SumOfHookLengthsSt000869The total of all hook lengths.
• SymmetricDiagonalHooksSt001525Cells on the main diagonal whose arm and leg are equal.
• WeightedSizeSt000185The sum of the parts weighted by their row, counting rows from 0.

Maps

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

From IntegerPartition

• TransposePartitionIntegerPartition → IntegerPartitioninvolutionBijectionGraded mapInvolutionMp00044The conjugate partition: column i of the Ferrers diagram becomes row i.

To IntegerPartition

• ConjugateAfterCycleTypePermutation → IntegerPartitionThe conjugate partition of the cycle type.
• CycleTypePermutation → IntegerPartitionSurjectionGraded mapMp00108The multiset of cycle lengths, as a partition.
• RskShapePermutation → IntegerPartitionSurjectionGraded mapMp00060The common shape of the RSK tableaux: the row lengths, as a partition of n.

37 more in FindStat, not implemented here

• Mp00021CorePartition → IntegerPartitionGraded mapBijectionto bounded partition
• Mp00022CorePartition → IntegerPartitionSurjectionto partition
• Mp00027DyckPath → IntegerPartitionInjectionto partition
• Mp00037Graphs → IntegerPartitionSurjectionGraded mapto partition of connected components
• Mp00040Composition → IntegerPartitionSurjectionGraded mapto partition
• Mp00042IntegerPartition → StandardTableauInjectionGraded mapinitial tableau
• Mp00043IntegerPartition → DyckPathInjectionto Dyck path
• Mp00045IntegerPartition → StandardTableauInjectionGraded mapreading tableau
• Mp00077SemistandardTableau → IntegerPartitionSurjectionGraded mapshape
• Mp00079SetPartition → IntegerPartitionSurjectionGraded mapshape
• Mp00083StandardTableau → IntegerPartitionSurjectionGraded mapshape
• Mp00095IntegerPartition → BinaryWordInjectionto binary word
• Mp00110Posets → IntegerPartitionSurjectionGraded mapGreene-Kleitman invariant
• Mp00166SignedPermutation → IntegerPartitionSurjectioneven cycle type
• Mp00169SignedPermutation → IntegerPartitionSurjectionodd cycle type
• Mp00179IntegerPartition → SkewPartitionInjectionGraded mapto skew partition
• Mp00182SkewPartition → IntegerPartitionSurjectionouter shape
• Mp00183SkewPartition → IntegerPartitionSurjectioninner shape
• Mp00186SkewPartition → IntegerPartitionSurjectionGraded mapdominating partition
• Mp00202IntegerPartition → IntegerPartitionSurjectionfirst row removal
• Mp00204Permutation → IntegerPartitionSurjectionGraded mapLLPS
• Mp00225SemistandardTableau → IntegerPartitionSurjectionweight
• Mp00230IntegerPartition → DyckPathInjectionparallelogram polyomino
• Mp00251Graphs → IntegerPartitionclique sizes
• Mp00275Graphs → IntegerPartitionSurjectionto edge-partition of connected components
• Mp00276Graphs → IntegerPartitionto edge-partition of biconnected components
• Mp00306Posets → IntegerPartitionrowmotion cycle type
• Mp00307Posets → IntegerPartitionpromotion cycle type
• Mp00308IntegerPartition → IntegerPartitionGraded mapBulgarian solitaire
• Mp00311PlanePartition → IntegerPartitionGraded mapto partition
• Mp00312IntegerPartition → IntegerPartitionBijectionGraded mapGlaisher-Franklin
• Mp00313IntegerPartition → IntegerPartitionBijectionGraded mapGlaisher-Franklin inverse
• Mp00317IntegerPartition → BinaryWordSurjectionodd parts
• Mp00321IntegerPartition → IntegerPartitionBijectionGraded mapInvolution2-conjugate
• Mp00322IntegerPartition → IntegerPartitionBijectionGraded mapLoehr-Warrington
• Mp00323IntegerPartition → IntegerPartitionBijectionGraded mapLoehr-Warrington inverse
• Mp00332Posets → IntegerPartitionSurjectionGraded mapJordan block partition

Examples

Basic #

$\binom{5}{2} = 10$

live ↗

In$\mathrm{Count}(\mathrm{PartitionsInBox}(2, 3))$Out$10$

At most 2 parts, each at most 2

live ↗

In$\mathrm{Take}(\mathrm{PartitionsInBox}(2, 2), 6)$Out$\bigl\lbrack{\operatorname{IntegerPartition}}([2, 2]), {\operatorname{IntegerPartition}}([2, 1]), {\operatorname{IntegerPartition}}([1, 1]), {\operatorname{IntegerPartition}}([2]), {\operatorname{IntegerPartition}}([1]), {\operatorname{IntegerPartition}}([])\bigr\rbrack$

About

Examples	2
Crosswalk	OEIS

See also: IntegerPartitions, PartitionsMaxPart, PartitionsIntoKParts