SortedPartition from IntegerComposition

The partition a composition's parts form, sorted largest first.

IntegerComposition → IntegerPartitionsurjection

Lossy: the order of the parts is forgotten, so it is not a constructor overload.

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

Live

Explore

Each element of IntegerComposition at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on IntegerPartition. Its distribution over the images of IntegerComposition at each size is computed, and compared with the 44 statistics on IntegerComposition: 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 Mp00040

to partition

Matched by value: our map agrees with FindStat's image on each of the 63 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

SurjectionGraded mapas FindStat records them, checked by FindStat's contributors.

Sends a composition to the partition obtained by sorting the entries.

Object	Image
[1]	[1]
[2]	[2]
[3]	[3]
[4]	[4]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00040 has the rest.