ReverseComplement from DyckPath

The path read backwards, with ups and downs swapped: its mirror image across the vertical.

DyckPath → DyckPathinvolution

FindStat's "reverse" for Dyck paths: reading the steps backwards alone would leave the word's ups and downs unbalanced, so each step is also swapped.

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

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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

Also from

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

FindStat Mp00028

reverse

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.

The reversal of a Dyck path.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00028 has the rest.