InverseFoata from Permutation

The permutation whose cycles are the stretches of the word between successive left-to-right maxima, each read as a cycle.

Permutation → Permutationbijectioninverse

FindStat's Mp00086, the inverse of Foata: a word's left-to-right maxima open its cycles.

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

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Each element of Permutation at a size, and its image, applied in your browser; sizes past 7 are not offered.

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Properties

Sizes 1 to

Transport a statistic

Pick a statistic on Permutation. Its distribution over the images of Permutation at each size is computed, and compared with the 323 statistics on Permutation: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic DescentsAscentsMajorIndexMinorIndexInversionsSignFixedPointsExcedancesWeakExcedancesAntiexcedancesDenertPeaksValleysLeftToRightMaximaLeftToRightMinimaRightToLeftMaximaRightToLeftMinimaFirstDescentFirstAscentLastDescentRunsDepthCyclicDescentsOccurrencesOf123OccurrencesOf132OccurrencesOf213StackSortableLongestRunLargestRunLengthCycleCountReflectionLengthLargestCycleLengthLongestCycleLengthDistinctCycleLengthsTwoCycleCountThreeCycleCountPermutationOrderPermutationSupportLongestIncreasingSubsequenceLongestDecreasingSubsequenceLeftOuterPeaksRightOuterPeaksOuterPeaksDoubleAscentsDoubleDescentsIsolatedDescentsBigAscentsBigDescentsDescentTopSumDescentBottomSumAscentTopSumAscentBottomSumComajorIndexNontrivialCyclesShortestCycleLengthConjugacyClassSizeWeakAntiexcedancesOccurrencesOf321OccurrencesOf312OccurrencesOf231NonLeftToRightMaximaExclusiveLeftToRightMaximaExclusiveRightToLeftMinimaNonRecordsNonInversionsInversionSumEvenInversionsOddInversionsInverseMajorIndexFirstEntryLastEntrySuccessionsAdjacenciesTwoRisesMaximalRiseSizeBigAscentsAfterPrependingZeroVeryBigAscentsVeryBigDescentsSmallExcedancesTwoExcedancesBigExcedancesDoubleExcedancesDoubleDeficienciesNestedExcedancesPureExcedancesGlobalAscentsGlobalDescentsDecompositionNumberCyclesOfLengthAtLeastThreeInversionsOfDistanceAtMostTwoNonInversionSumTotalDisplacementDescentDifferenceSumShiftedMajorIndexRecoilsDescentsOrRecoilsLeftToRightMaximaPositionSumArithmeticProgressionsOfLengthTwoOccurrencesOf1324OccurrencesOf2143OccurrencesOf4231OccurrencesOf2413OccurrencesOf1432OccurrencesOf21Dash3OccurrencesOf13Dash2OccurrencesOf12Dash3OccurrencesOf31Dash2OccurrencesOf23Dash1OccurrencesOf32Dash1OccurrencesOfInitial213OccurrencesOfInitial231OccurrencesOfInitial312OccurrencesOfInitial321OccurrencesOfInitial132OccurrencesOfInitial123OccurrencesOfInitial12Dash3OccurrencesOfInitial2Dash13OccurrencesOfInitial1Dash23OccurrencesOfConsecutive132OccurrencesOf3241OccurrencesOf4213OccurrencesOf123Or132OccurrencesOf213Or321OccurrencesOf132Or321OccurrencesOf132Or213Or321OccurrencesOf3241Or4231OccurrencesOf4132Or4231OccurrencesOf3412Or4231OccurrencesOf2143Or4213SmallWeakExcedancesNonSmallExcedancesNonSmallWeakExcedancesCyclicSmallExcedancesCyclicSmallWeakExcedancesNonCyclicSmallExcedancesNonCyclicSmallWeakExcedancesOccurrencesOf132Or231OccurrencesOf132Or213OccurrencesOf132Or312OccurrencesOf123Or231OccurrencesOf123Or213OccurrencesOf123Or321OccurrencesOf123Or312OccurrencesOf231Or312OccurrencesOf213Or312OccurrencesOf213Or231OccurrencesOf231Or321OccurrencesOf312Or321OccurrencesOf321Or3412OccurrencesOf52341OccurrencesOf41352Or42351Or51342Or52341ThreeExcedancesThreeRisesBigDeficienciesDescentsOfDistanceTwoAscentsOfDistanceTwoDoubleSuccessionsDescentsPlusRecoilsDescentsAndRecoilsOddDescentsEvenDescentsFinalRisesMidPointsOfOccurrencesOf321MidPointsOfOccurrencesOf123LeftToRightMaximaOrRightToLeftMinimaMajorIndexMinusInversionsInversionsPlusMajorIndexMajorIndexMinusExcedancesMaximalDisplacementValleysIncludingBoundaryMaximalExcedanceAdjacentCyclesStrongFixedPointsSortingIndexLoadCycleDescentsDescentsAfterZeroingFixedPointsMajorIndexAfterZeroingFixedPointsElizaldePakRankDisorderCyclesOfLengthAtMostTwoCyclesOfLengthAtMostThreeInversionsOfDistanceAtMostThreeFirstDirectSummandSizeLargestDirectSummandSizeStretchingPairsInversionIndexTotalSquaredDisplacementLeftToRightMaximaXorRightToLeftMinimaMaximalSuccessiveDifferenceMajorIndexPlusInverseMajorIndexTierRaftsWeakExcedanceMidPointsOfOccurrencesOf321NonWeakExcedanceMidPointsOfOccurrencesOf321TwoShiftedMajorIndexThreeShiftedMajorIndexPinnacleSumAugmentedDoubleAscentsAlternatingRunsUpDownRunsNonInitialLeftToRightMinimaRightFloatsRightRopesRightTethersMinimalJumpMaximalJumpCyclicMaximalJumpNeighboringNumberDistinctDiagonalSumsMaximalDiagonalOccupancyDescentsOfDistanceThreeRepeatedLehmerCodeEntriesAdjacenciesWithZeroAppendedSmallestFixedPointLargestWeakExcedancePositionFixedPointsBeforeLargestMovedPointAscentTopsWithSmallerValuesBeforeDescentsOfTypeTwoLongestSuccessionRunMaximalModularDisplacementAdjacentTranspositionsCyclicallyAdjacentTranspositionsBondsCyclicBondsCrossingsNestingsAlignmentsCyclicValleysCyclePeaksNonOverlappingDescentsVisibleInversionsInvisibleInversionsPrefixSumProductOneBoxPatternsMaximumOfDescentsAndInverseDescentsInvisibleDescentsVisibleDescentsGreaterNeighborOfMaximumInversionsOfThirdEntryInversionsOfSecondEntrySumOfDistancesToGreaterElementNormalizedSumOfDistancesToGreaterElementHalfParityMismatchesDescentWeightMinusInversionsReducedWordSupportPermutationCosineMakMadMaklBabsonSteingrimssonStatisticUlamDistanceEvenWeakDeficienciesAndOddWeakExcedancesVerticalSeparatorsCyclicAlignmentsCyclicMisalignmentsCyclicCrossingsRestrictedNonInversionsBetweenExcedancesInversionsBetweenExcedancesWithLinkedGreaterRestrictedNonInversionsBetweenExcedancesWithLinkedRightmostInversionsBetweenExcedancesAndFixedPointsUpperMiddleEntryLowerMiddleEntryConvexityProperBlocksHertzsprungOccurrencesOf132PositionsOfLetterOneIn123PositionsOfLetterOneIn132PositionsOfLetterTwoIn213PositionsOfLetterThreeIn132OccurrencesOf14Dash2Dash3Or14Dash3Dash2InversionsOfCyclicEmbeddingShareshianWachsAdmissibleInversionsAdmissibleInversionsPlusDescentsAdmissibleInversionsLargestGreatestCommonDivisorOfPositionAndValueMaximalCycleDifferenceLongestKOneTwoPatternSmallestLeafLabelLongestIncreasingSubsequenceCountPrimeColumnSumsPrimeProductsMinusOnePrimeProductsPlusOneLongestArithmeticProgressionDiaconisGrahamDefectInterpolatingPolynomialDegreeMinimalCycleDifferenceNonFixedPointsLargestSearchTreeLeafLabelNormalizedLeafLabelSumDescentsMinusOneUnlessFirstEntryOnePermutationBreadthBoxedOccurrencesOf123BoxedOccurrencesOf132BinStatisticNonNegativeWalkStepsCycleMinimumExceedancesDeficienciesSeparatorsOverlapSetSizeExteriorLengthSimsunDoubleDescentsSymmetryClassSizePermutationGenusPermutationBounceLexicographicRankNonAttackingNeighborsStaircaseSizeOfCodeBruhatLowerCoversTailPairsOfOccurrencesOf321StandardizedBiAlternatingInversionNumberMinimalInversionsEitherDirectionBabsonSteingrimssonStatisticPrimeBabsonSteingrimssonStatisticDoublePrimeHagHalfPermanentOfIdentityPlusPermutationHookFactorizationPrefixLengthHookFactorInversionsPermutationTreeHeightFixedPointMajorIndexRecursiveEndMaximumCountRecursiveStartMinimumCountSmallerChildPathLeafLabelScaledReciprocalPositionValueSum Sizes 1 to

FindStat Mp00086

first fundamental transformation

Matched by value: our map agrees with FindStat's image on each of the 873 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.

Return the permutation whose cycles are the subsequences between successive left to right maxima.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00086 has the rest.