GlaisherFranklin from IntegerPartition

The Glaisher-Franklin bijection: even part sizes halved become repeated parts, and repeated odd parts merge into their binary multiples.

IntegerPartition → IntegerPartitionbijectioninverse

FindStat's Mp00312. It sends the set of even part sizes, each halved, to the set of repeated part sizes.

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

Live

Explore

Each element of IntegerPartition 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 IntegerPartition at each size is computed, and compared with the 165 statistics on IntegerPartition: 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 Mp00312

Glaisher-Franklin

Matched by value: our map agrees with FindStat's image on each of the 29 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

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

The Glaisher-Franklin bijection on integer partitions.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00312 has the rest.