Permutation from CycleDecomposition

The permutation a cycle decomposition describes.

CycleDecomposition → Permutationinverse

Declines a decomposition that isn't canonical: a point missing or repeated, a cycle not starting at its least point, or cycles out of order.

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

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Properties

Sizes 1 to

Transport a statistic

Pick a statistic on Permutation. Its distribution over the images of CycleDecomposition at each size is computed, and compared with the 0 statistics on CycleDecomposition: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic DescentsAscentsMajorIndexMinorIndexInversionsSignFixedPointsExcedancesWeakExcedancesAntiexcedancesDenertPeaksValleysLeftToRightMaximaLeftToRightMinimaRightToLeftMaximaRightToLeftMinimaFirstDescentFirstAscentLastDescentRunsDepthCyclicDescentsOccurrencesOf123OccurrencesOf132OccurrencesOf213StackSortableLongestRunLargestRunLengthCycleCountReflectionLengthLargestCycleLengthLongestCycleLengthDistinctCycleLengthsTwoCycleCountThreeCycleCountPermutationOrderPermutationSupportLongestIncreasingSubsequenceLongestDecreasingSubsequenceLeftOuterPeaksRightOuterPeaksOuterPeaksDoubleAscentsDoubleDescentsIsolatedDescentsBigAscentsBigDescentsDescentTopSumDescentBottomSumAscentTopSumAscentBottomSumComajorIndexNontrivialCyclesShortestCycleLengthConjugacyClassSizeWeakAntiexcedancesOccurrencesOf321OccurrencesOf312OccurrencesOf231NonLeftToRightMaximaExclusiveLeftToRightMaximaExclusiveRightToLeftMinimaNonRecordsNonInversionsInversionSumEvenInversionsOddInversionsInverseMajorIndexFirstEntryLastEntrySuccessionsAdjacenciesTwoRisesMaximalRiseSizeBigAscentsAfterPrependingZeroVeryBigAscentsVeryBigDescentsSmallExcedancesTwoExcedancesBigExcedancesDoubleExcedancesDoubleDeficienciesNestedExcedancesPureExcedancesGlobalAscentsGlobalDescentsDecompositionNumberCyclesOfLengthAtLeastThreeInversionsOfDistanceAtMostTwoNonInversionSumTotalDisplacementDescentDifferenceSumShiftedMajorIndexRecoilsDescentsOrRecoilsLeftToRightMaximaPositionSumArithmeticProgressionsOfLengthTwoOccurrencesOf1324OccurrencesOf2143OccurrencesOf4231OccurrencesOf2413OccurrencesOf1432OccurrencesOf21Dash3OccurrencesOf13Dash2OccurrencesOf12Dash3OccurrencesOf31Dash2OccurrencesOf23Dash1OccurrencesOf32Dash1OccurrencesOfInitial213OccurrencesOfInitial231OccurrencesOfInitial312OccurrencesOfInitial321OccurrencesOfInitial132OccurrencesOfInitial123OccurrencesOfInitial12Dash3OccurrencesOfInitial2Dash13OccurrencesOfInitial1Dash23OccurrencesOfConsecutive132OccurrencesOf3241OccurrencesOf4213OccurrencesOf123Or132OccurrencesOf213Or321OccurrencesOf132Or321OccurrencesOf132Or213Or321OccurrencesOf3241Or4231OccurrencesOf4132Or4231OccurrencesOf3412Or4231OccurrencesOf2143Or4213SmallWeakExcedancesNonSmallExcedancesNonSmallWeakExcedancesCyclicSmallExcedancesCyclicSmallWeakExcedancesNonCyclicSmallExcedancesNonCyclicSmallWeakExcedancesOccurrencesOf132Or231OccurrencesOf132Or213OccurrencesOf132Or312OccurrencesOf123Or231OccurrencesOf123Or213OccurrencesOf123Or321OccurrencesOf123Or312OccurrencesOf231Or312OccurrencesOf213Or312OccurrencesOf213Or231OccurrencesOf231Or321OccurrencesOf312Or321OccurrencesOf321Or3412OccurrencesOf52341OccurrencesOf41352Or42351Or51342Or52341ThreeExcedancesThreeRisesBigDeficienciesDescentsOfDistanceTwoAscentsOfDistanceTwoDoubleSuccessionsDescentsPlusRecoilsDescentsAndRecoilsOddDescentsEvenDescentsFinalRisesMidPointsOfOccurrencesOf321MidPointsOfOccurrencesOf123LeftToRightMaximaOrRightToLeftMinimaMajorIndexMinusInversionsInversionsPlusMajorIndexMajorIndexMinusExcedancesMaximalDisplacementValleysIncludingBoundaryMaximalExcedanceAdjacentCyclesStrongFixedPointsSortingIndexLoadCycleDescentsDescentsAfterZeroingFixedPointsMajorIndexAfterZeroingFixedPointsElizaldePakRankDisorderCyclesOfLengthAtMostTwoCyclesOfLengthAtMostThreeInversionsOfDistanceAtMostThreeFirstDirectSummandSizeLargestDirectSummandSizeStretchingPairsInversionIndexTotalSquaredDisplacementLeftToRightMaximaXorRightToLeftMinimaMaximalSuccessiveDifferenceMajorIndexPlusInverseMajorIndexTierRaftsWeakExcedanceMidPointsOfOccurrencesOf321NonWeakExcedanceMidPointsOfOccurrencesOf321TwoShiftedMajorIndexThreeShiftedMajorIndexPinnacleSumAugmentedDoubleAscentsAlternatingRunsUpDownRunsNonInitialLeftToRightMinimaRightFloatsRightRopesRightTethersMinimalJumpMaximalJumpCyclicMaximalJumpNeighboringNumberDistinctDiagonalSumsMaximalDiagonalOccupancyDescentsOfDistanceThreeRepeatedLehmerCodeEntriesAdjacenciesWithZeroAppendedSmallestFixedPointLargestWeakExcedancePositionFixedPointsBeforeLargestMovedPointAscentTopsWithSmallerValuesBeforeDescentsOfTypeTwoLongestSuccessionRunMaximalModularDisplacementAdjacentTranspositionsCyclicallyAdjacentTranspositionsBondsCyclicBondsCrossingsNestingsAlignmentsCyclicValleysCyclePeaksNonOverlappingDescentsVisibleInversionsInvisibleInversionsPrefixSumProductOneBoxPatternsMaximumOfDescentsAndInverseDescentsInvisibleDescentsVisibleDescentsGreaterNeighborOfMaximumInversionsOfThirdEntryInversionsOfSecondEntrySumOfDistancesToGreaterElementNormalizedSumOfDistancesToGreaterElementHalfParityMismatchesDescentWeightMinusInversionsReducedWordSupportPermutationCosineMakMadMaklBabsonSteingrimssonStatisticUlamDistanceEvenWeakDeficienciesAndOddWeakExcedancesVerticalSeparatorsCyclicAlignmentsCyclicMisalignmentsCyclicCrossingsRestrictedNonInversionsBetweenExcedancesInversionsBetweenExcedancesWithLinkedGreaterRestrictedNonInversionsBetweenExcedancesWithLinkedRightmostInversionsBetweenExcedancesAndFixedPointsUpperMiddleEntryLowerMiddleEntryConvexityProperBlocksHertzsprungOccurrencesOf132PositionsOfLetterOneIn123PositionsOfLetterOneIn132PositionsOfLetterTwoIn213PositionsOfLetterThreeIn132OccurrencesOf14Dash2Dash3Or14Dash3Dash2InversionsOfCyclicEmbeddingShareshianWachsAdmissibleInversionsAdmissibleInversionsPlusDescentsAdmissibleInversionsLargestGreatestCommonDivisorOfPositionAndValueMaximalCycleDifferenceLongestKOneTwoPatternSmallestLeafLabelLongestIncreasingSubsequenceCountPrimeColumnSumsPrimeProductsMinusOnePrimeProductsPlusOneLongestArithmeticProgressionDiaconisGrahamDefectInterpolatingPolynomialDegreeMinimalCycleDifferenceNonFixedPointsLargestSearchTreeLeafLabelNormalizedLeafLabelSumDescentsMinusOneUnlessFirstEntryOnePermutationBreadthBoxedOccurrencesOf123BoxedOccurrencesOf132BinStatisticNonNegativeWalkStepsCycleMinimumExceedancesDeficienciesSeparatorsOverlapSetSizeExteriorLengthSimsunDoubleDescentsSymmetryClassSizePermutationGenusPermutationBounceLexicographicRankNonAttackingNeighborsStaircaseSizeOfCodeBruhatLowerCoversTailPairsOfOccurrencesOf321StandardizedBiAlternatingInversionNumberMinimalInversionsEitherDirectionBabsonSteingrimssonStatisticPrimeBabsonSteingrimssonStatisticDoublePrimeHagHalfPermanentOfIdentityPlusPermutationHookFactorizationPrefixLengthHookFactorInversionsPermutationTreeHeightFixedPointMajorIndexRecursiveEndMaximumCountRecursiveStartMinimumCountSmallerChildPathLeafLabelScaledReciprocalPositionValueSum Sizes 1 to

Also from

The same name out of other carriers, each its own implementation: SubexcedantSeq.