CutOutPartition from DyckPath

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.

DyckPath → IntegerPartition

FindStat's Mp00027, "to partition". Injective: its image is the partitions inside the staircase (n-1, ..., 1).

Called as CombinatorialMap(x, "CutOutPartition"). The head's page has its examples.

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Each element of DyckPath at a size, and its image, applied in your browser; sizes past 7 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on IntegerPartition. Its distribution over the images of DyckPath at each size is computed, and compared with the 171 statistics on DyckPath: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic LargestPartMultiplicityOfLargestPartDistinctPartsEvenPartsOddPartsPartsEqualOnePartsAtLeastTwoConjugateOddPartsConjugateDistinctPartsDurfeeSquareArmOfFirstCellLegOfFirstCellCornersHalfPerimeterIsSelfConjugateSumOfHookLengthsHookProductStandardTableauCountDysonRankCrankHookLengthOfFirstCellHookLengthOfLastDiagonalCellDiagonalHookLengthProductSymmetricDiagonalHooksCellsOutsideDurfeeSquareFirstRowHookLengthSumHookLengthSquareSumCohookLengthProductDiagonalInversionsDiagonalInversionDefectCellsWithEqualArmAndLegLargestRepeatedPartMultiplicityOfSmallestPartFlooredHalfMultiplicitySumDistinctRepeatedPartsSingletonPartsDistinctMultiplicitiesLargestMultiplicityPartsEqualToMultiplicityDistinctPartsEqualToMultiplicityWeightedSizeAddableCellsPartsExceedingOnesContentSumLargestStaircaseSideContainingSquareSideCellsOutsideLargestStaircaseAftSmallestMissingPartPairwiseProductSumPartRangeContainingRectangleShortSideMaximalRectangleHalfPerimeterLargestShiftedPartRemovableHorizontalDominoesCellsOutsideFirstRowCellsOutsideFirstHookRowCellPairsCellsWithZeroLegAndOddArmCornerMajorIndexPartProductPartSquareSumPartFactorialProductMultiplicityFactorialProductSumOfEvenPartsSumOfOddPartsSumOfPartsAtLeastTwoGreatestCommonDivisorOfPartsLeastCommonMultipleOfPartsPairwiseGreatestCommonDivisorSumLargestEvenPartEvenOddCrankOddPartsBelowLargestEvenPartDistinctEvenPartsBerkovichGarvanRankPartsEqualToNextPartLargestMissingPartSmallestNonRepeatedIntegerDistinctSubsetSumsDistinctEvenSubsetSumsDistinctOddSubsetSumsPartsNotCongruentToTwoModuloThreePartsNotCongruentToZeroModuloThreePartsNotCongruentToOneModuloThreePartsEqualTwoHalfSumOfEvenPartsHalfLargestEvenPartDistinctOddPartsBelowLargestEvenPartPartSquareProductMultinomialOfPartsExponensConsonantiaeCyclicSievingPolynomialDegreeGiniIndexInterlacingPartitionCountMinimalSubpartitionSizeDifferenceAlladiWeightSecondAlladiWeightCentralizerSizeConjugacyClassSizeSetPartitionsWithBlockSizesCountKrewerasNumberInvariantSubsetCountInvariantTwoElementSubsetCountInvariantThreeElementSubsetCountEdgeOrbitCountElementarySymmetricFunctionAtOneSubsetGreatestCommonDivisorSumSemistandardTableauCountWithEntriesUpToTwoSemistandardTableauCountWithEntriesUpToThreeSemistandardTableauCountWithEntriesUpToFourSemistandardTableauCountWithEntriesUpToLengthSemistandardTableauCountWithEntriesUpToSizePlancherelCountCubedStandardTableauCountShiftedStandardTableauCountTwiceMeanMajorIndexOfStandardTableauxTwelveTimesMajorIndexVarianceOfStandardTableauxPairsOfDistinctPartsDiagonalArmSumRimHooksRemovedToTwoCoreRimHooksRemovedToThreeCoreMultipartitionCountPartRearrangementCountLagrangeInversionCoefficientLogarithmicDerivativeCoefficientStarTranspositionFactorizationCountTransitiveMonotoneFactorizationCountSymplecticDimensionRankTwoSymplecticDimensionRankThreeBorderSkewStandardTableauCountSubpartitionCountSameLengthSubpartitionCountYoungLatticeCoveringRelationCountPartCountLengthCrankSeaweedAlgebraDimensionCartanDeterminantRookPlacementCountMaximalRookPlacementCountShorterSideRookPlacementCountNecklaceCountBraceletCountBulgarianSolitairePreimageCountInvariantOrientedCycleCountSmirnovWordCountStandardTableauCountWithMajorIndexCongruentToOneModuloSizeUpperCoverCountInDominanceOrderLowerCoverCountInDominanceOrderInvariantGraphCountCompleteHomogeneousSymmetricFunctionAtOnePowerSumSymmetricFunctionAtOneCubeRootCountDominatedPartitionCountBulgarianSolitaireOrbitSizeBulgarianSolitairePreperiodPartitionSizeCouplesAreForeverGrundyValueRowRemovalGrundyValueOrderedRefinementCountRefinementCountCoarseningCountSingletonRefinementCountHurwitzNumberMonotoneHurwitzNumberNonDominatingSubmultisetCount Sizes 1 to

FindStat Mp00027

to partition

Matched by value: our map agrees with FindStat's image on each of the 196 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

Injectionas FindStat records them, checked by FindStat's contributors.

The cut-out partition of a Dyck path.

Object	Image
[1,0]	[]
[1,0,1,0]	[1]
[1,1,0,0]	[]
[1,0,1,0,1,0]	[2,1]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00027 has the rest.