SwapReturnsAndLastDescent from DyckPath

An involution exchanging the number of returns with the length of the last descent.

DyckPath → DyckPath

FindStat's Mp00118, the specialization to Dyck paths of the map of Elizalde and Rubey. It keeps the number of ups before each down that is neither a return nor in the last descent.

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

Live

Explore

Each element of DyckPath 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 DyckPath. 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 HeightPeaksValleysDoubleRisesReturnsTouchPointCountInteriorReturnsHillsInitialRiseMajorIndexAreaCoareaLongestAscentLongestDescentDinvBounceBounceCountDinvDeficitBounceDeficitFirstComponentSemilengthDownStepHeightProductValleysAboveAxisOccurrencesOfDDUOccurrencesOfUUUOccurrencesOfUDUPeakHeightSumPeakHeightExcessPeakHeightSquareSumPeakHeightProductLastUpStepPositionFirstDownStepPositionLastDoubleRisePositionRisesOfLengthOneMinimalRiseRisesOfLengthAtLeastTwoUpStepsAtOddPositionsUpStepsAfterLastDoubleRiseDoubleRisePositionSumDoubleFallPositionSumMacMahonEqualIndexLogarithmicHeightFirstMaximalPeakPositionGlobalMaximaLargestAreaSequenceEntryUpStepsAtEvenPositionsUpperInteractionsPyramidWeightCenteredTunnelsLeftTunnelsNestedCenteredTunnelPairsNestedLeftTunnelPairsValleyTunnelSemilengthSumCenteredMultitunnelsRisesOfLengthTwoRisesOfLengthThreeRisesOfLengthAtLeastThreeRunsOfLengthOneRisesOfLengthOneAtOddHeightPeakRectangleAreaSumSkewHookPositionSumDownStepMajorIndexUpStepMajorIndexLongestZigzagBicoloredMotzkinHeightBicoloredMotzkinLevelStepsMinimalValleyHeightHillsOfSizeTwoHillsOfSizeThreeMinimalPeakHeightPeakLeftToRightMaximaWeakPeakLeftToRightMaximaBoxWeightMiddleNonRunHeightSymmetryDegreeParkingFunctionCountValleyPositionSumInteractionDifferenceNonFinalConstantRunsOfLengthAtLeastTwoLongTunnelsHalfMacMahonEqualIndexBoxesBelowOnePeakBoxesBelowAtLeastTwoPeaksDoubleStepVisiblePointsAreaAboveBouncePathWeaklyBelowDyckPathCountProperlyBelowDyckPathCountWeaklyAboveDyckPathCountProperlyAboveDyckPathCountWeaklyAboveLatticePathCountNorthStepOrderingCountThreeCatalanPathCountParallelogramPolyominoAreaParallelogramPolyominoNormalizedAreaParallelogramPolyominoMinimalColumnHeightParallelogramPolyominoMaximalColumnHeightParallelogramPolyominoInnerCornersParallelogramPolyominoUpperInnerCornersSetValuedTableauNaturalDescentsSetValuedTableauComajorIndexAdjointInvariantDescentsDiagonalStandardMonomialDegreeNoncrossingMergeableSingletonPairsNakayamaGlobalDimensionFirstSimpleProjectiveDimensionSimpleProjectiveDimensionSumSimpleProjectiveDimensionProductSimplesOfMaximalProjectiveDimensionSimplesOfProjectiveDimensionTwoSimplesOfProjectiveDimensionThreeSimplesOfProjectiveDimensionFourSimplesOfProjectiveDimensionAtMostTwoSimplesOfProjectiveDimensionAtMostThreeSimplesOfProjectiveDimensionAtMostFourSimplesOfEvenProjectiveDimensionSimplesOfMaximalEvenProjectiveDimensionHalfEvenSimpleProjectiveDimensionSumFirstSimpleOfMaximalProjectiveDimensionLastSimpleOfMaximalProjectiveDimensionPeaksMinusNakayamaGlobalDimensionInjectiveProjectiveDimensionSumInjectivesOfProjectiveDimensionOneInjectivesOfProjectiveDimensionAtMostOneInjectivesOfProjectiveDimensionAtMostTwoInjectivesOfMaximalProjectiveDimensionInjectivesOfProjectiveDimensionOneBelowMaximalInjectivesOfEvenProjectiveDimensionModulesOfMaximalProjectiveDimensionModulesOfProjectiveDimensionAtMostOneInjectivesOfProjectiveDimensionTwoModulesOfProjectiveDimensionTwoTwoDimensionalModulesOfProjectiveDimensionOneThreeDimensionalModulesOfProjectiveDimensionOneSecondToLastInjectiveProjectiveDimensionSimpleSummandsOfSecondRadicalLayerNakayamaDominantDimensionNakayamaGlobalMinusDominantDimensionPeaksMinusNakayamaDominantDimensionInjectivesOfCodominantDimensionEqualToGlobalDimensionInjectivesOfCodominantDimensionAtLeastTwoNonProjectiveInjectiveCodominantDimensionSumProjectiveMinusCodominantDimensionSumSimplesOfNonMaximalProjectiveDimensionNonProjectiveInjectiveProjectiveDimensionSpreadProjectivesOfMaximalInjectiveDimensionSimplesOfEqualProjectiveAndInjectiveDimensionModulesOfMaximalProjectiveAndInjectiveDimensionModulesOfProjectiveAndInjectiveDimensionAtMostOneModulesOfProjectiveOrInjectiveDimensionAtMostOneModulesOfProjectiveAndInjectiveDimensionAtLeastTwoMaximalSimpleProjectiveAndInjectiveDimensionSumMaximalModuleProjectiveAndInjectiveDimensionSumSimplesOfSmallInjectiveDimensionOrPositiveDominantDimensionSimplesOfDominantDimensionEqualToGlobalDimensionSimplesOfDominantDimensionAtLeastThreeMaximalSimpleDominantDimensionUnitIntervalGraphTwinVerticesNonProjectiveInjectiveToNonInjectiveProjectiveHomDimensionDualRegularModuleFirstExtensionDimensionRadicalEndomorphismDimensionRadicalSelfExtensionDimensionRadicalToSquaredRadicalExtensionDimensionSocleToRadicalExtensionDimensionRegularModuloSocleToRadicalExtensionDimensionMaximalFirstExtensionWithRegularModuleModulesOfVanishingFirstExtensionWithRegularModuleSimplesOfVanishingSecondExtensionWithRegularModuleMaximalSecondExtensionOfSimplesWithRegularModuleSimplesOfTwoDimensionalSecondExtensionWithRegularModuleInjectivesOfNonVanishingSecondExtensionWithRegularModuleInjectivesOfNonMaximalSocleProjectiveDimensionMinusProjectiveInjectivesSocleSimplesOfVanishingFirstExtensionWithRegularModule Sizes 1 to

FindStat Mp00118

swap returns and last descent

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).

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

Return a Dyck path with number of returns and length of the last descent interchanged.

Object	Image
[1,0]	[1,0]
[1,0,1,0]	[1,1,0,0]
[1,1,0,0]	[1,0,1,0]
[1,0,1,0,1,0]	[1,1,1,0,0,0]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00118 has the rest.