Avoiding321Permutation from DyckPath

The 321-avoiding permutation whose RSK pair is the pair of two-row tableaux read off the first and last halves of the path.

DyckPath → Permutation

FindStat's Mp00024. It sends centered tunnels to fixed points and the semilength plus the height of the middle point to twice the longest increasing subsequence. Injective.

Called as CombinatorialMap(x, "Avoiding321Permutation"). The head's page has its examples.

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Sizes 1 to

Transport a statistic

Pick a statistic on Permutation. Its distribution over the images of DyckPath at each size is computed, and compared with the 171 statistics on DyckPath: 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 Mp00024

to 321-avoiding permutation

Matched by value: our map agrees with FindStat's image on each of the 196 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

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

Sends a Dyck path to a 321-avoiding permutation.

Object	Image
[1,0]	[1]
[1,0,1,0]	[2,1]
[1,1,0,0]	[1,2]
[1,0,1,0,1,0]	[2,1,3]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00024 has the rest.