SupportDyckPath from ParkingFunction

The Dyck path whose area sequence is j minus the j-th smallest entry: the parking function's support.

ParkingFunction → DyckPath

FindStat's Mp00056, Sage's to_area_sequence. A surjection: the preimage of a path is its parking functions, the labelings of its up steps.

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

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Each element of ParkingFunction at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on DyckPath. Its distribution over the images of ParkingFunction at each size is computed, and compared with the 9 statistics on ParkingFunction: 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 Mp00056

to Dyck path

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

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

Sends a parking function to its support Dyck word.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00056 has the rest.