TableauShape from StandardTableau

The shape of a standard tableau: its row lengths.

StandardTableau → IntegerPartitionsurjection

Lossy: the entries are forgotten, so it is not a constructor overload.

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

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Each element of StandardTableau 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 StandardTableau at each size is computed, and compared with the 25 statistics on StandardTableau: 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 Mp00083

shape

Matched by value: our map agrees with FindStat's image on each of the 119 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 tableau to its shape.

Object	Image
[[1]]	[1]
[[1,2]]	[2]
[[1,2,3]]	[3]
[[1],[2]]	[1,1]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00083 has the rest.