ParkingFunction
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
Catalogued in the enumeratio database; not yet written up here.
Wikipedia
Parking functionvia ParkingFunctionsOEISA000272via ParkingFunctionsA000108via NonDecreasingParkingFunctionsSageMathsage.combinat.parking_functions.ParkingFunctions(n)via ParkingFunctionsParkingFunction(...)Catalogued in the enumeratio database; not yet written up here.Typed signatures
| Type | Package | Applies |
|---|---|---|
(list<integer>) -> parking_function | combinatorics | always |
Statistics
On ParkingFunction, called as CombinatorialStat(x, "Name").
- CenterSizeSt001937The length of the longest subsequence a(1), ..., a(k) with a(i) <= i.
- CriticalLeftToRightMaximaSt000942The entries p that are left to right maxima with exactly p - 1 entries smaller and n - p larger.
- DeficienciesSt001905Positions i with p(i) < i.
- InitialIncreasingSegmentLengthSt001904The length of the initial strictly increasing segment.
- LuckyCarsSt000135The cars that park in the spot they prefer.
- MaximalParkingDisplacementSt000943The most spots any car drives past its preferred spot.
- SumLessLengthSt000540The sum of the preferences less their number.
- TiesSt001947Positions i with p(i) = p(i+1).
- TotalParkingDisplacementSt000188The sum, over the cars, of the spots they drive past: the sum of i - p(i).
Maps
Into and out of ParkingFunction, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.
From ParkingFunction
- CarPermutationParkingFunction → PermutationSurjectionGraded mapMp00053Spot j holds the car that parks there, each car taking the first free spot at or after its preference.
- CyclicShiftParkingFunction → ParkingFunctioninverseBijectionGraded mapMp00293Rotate the word one place to the left: the first entry moves to the end.
- InverseCyclicShiftParkingFunction → ParkingFunctioninverseBijectionGraded mapMp00304Rotate the word one place to the right: the last entry moves to the front.
- LabelingPermutationParkingFunction → PermutationSurjectionGraded mapMp00055The labeling of the parking function's support Dyck path: the positions of the entries, smallest entry first and equal entries left to right.
- ReverseParkingFunction → ParkingFunctioninvolutionBijectionGraded mapInvolutionMp00320The word read backwards.
- SupportDyckPathParkingFunction → DyckPathSurjectionGraded mapMp00056The Dyck path whose area sequence is j minus the j-th smallest entry: the parking function's support.
13 more in FindStat, not implemented here
- Mp00052ParkingFunction → ParkingFunctionIdempotentGraded mapto non-decreasing parking function
- Mp00054ParkingFunction → IntegerCompositionSurjectionGraded mapto inverse des composition
- Mp00057ParkingFunction → IntegerCompositionSurjectionGraded mapto touch composition
- Mp00290ParkingFunction → SetCompositionSurjectionGraded mapto ordered set partition
- Mp00297ParkingFunction → OrderedTreeSurjectionGraded mapordered tree
- Mp00298ParkingFunction → ParkingFunctionBijectionGraded maprepeats to leading
- Mp00299ParkingFunction → ParkingFunctionBijectionGraded mapones to leading
- Mp00300ParkingFunction → ParkingFunctionBijectionGraded mapleading to ones
- Mp00301ParkingFunction → ParkingFunctionBijectionGraded mapleading to repeats
- Mp00302ParkingFunction → SemistandardTableauGraded mapinsertion tableau
- Mp00303ParkingFunction → ParkingFunctionBijectionGraded mapInvolutioncomplement
- Mp00305Permutation → ParkingFunctionInjectionGraded mapparking function
- Mp00319ParkingFunction → IntegerCompositionInjectionto composition