PositiveCycleType from SignedPermutation

The lengths of the positive cycles, largest first: those of the underlying permutation with an even number of negative entries.

SignedPermutation → IntegerPartition

FindStat's Mp00166, "even cycle type". Geck and Pfeiffer's positive cycles.

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

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Sizes 1 to

Transport a statistic

Pick a statistic on IntegerPartition. Its distribution over the images of SignedPermutation at each size is computed, and compared with the 18 statistics on SignedPermutation: 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 Mp00166

even cycle type

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

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

The partition corresponding to the even cycles.

Object	Image
[1]	[1]
[-1]	[]
[1,2]	[1,1]
[2,1]	[2]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00166 has the rest.