ElizaldeDeutsch from DyckPath

Reading the steps in the order first, last, second, second to last, and so on, a step of the image is an up step when its matching step has not yet been read.

DyckPath → DyckPath

FindStat's Mp00122, the bijection of Elizalde and Deutsch.

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

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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 Mp00122

Elizalde-Deutsch bijection

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

The Elizalde-Deutsch bijection on Dyck paths.

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]	[1,1,0,0,1,0]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00122 has the rest.