NonDecreasingParkingFunctions
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
Catalogued in the enumeratio database, with crosswalk rows in oeis; not yet written up here.
OEIS
A000108NonDecreasingParkingFunctions(...)Catalogued in the enumeratio database, with crosswalk rows in oeis; not yet written up here.Typed signatures
| Type | Package | Applies |
|---|---|---|
(integer<0..>) -> indexed_collection<parking_function> | combinatorics | always |
Definition
Epsil, as the implementation runs it.
count(_n)
_tables[_n * (_n + 2) + 1 + 1]unrank(_n, _r)
Drop(Fold(Function(Apply(
(up: list<integer>) => Join(
[up[1], up[2]],
Drop(us, 2),
[up[2]],
),
Fold(Function(
([uq[1], ua] if uq[1] < _tables[(_n -
ui) * (_n +
2) +
ua + 1] else [uq[1] -
_tables[(_n - ui) *
(_n + 2) + ua + 1], 0]) if uq[2] == 0 else uq,
uq,
ua,
), [us[1], 0], Range(us[2], ui, 1)),
), us, ui), [_r, 1], Range(1, _n, 1)), 2)rank(_n, _x)
Fold(Function([rs[1] + Fold(
Function(rt +
_tables[(_n - ri) * (_n + 2) + ra + 1], rt, ra),
0,
Range(rs[2], _x[ri] - 1, 1),
), _x[ri]], rs, ri), [0, 1], Range(1, _n, 1))[1]valid(_n, _x)
Length(_x) == _n && Fold(
Function(ok_i && 1 <= _x[i] && _x[i] <= i &&
(True if i == 1 else _x[i - 1] <=
_x[i]), ok_i, i),
True,
Range(1, Length(_x), 1),
)tables(_n)
Fold(
Function(Join(
pf_t,
Fold(
Function(
ReplaceAt(
pf_row,
pf_c + 1,
Fold(
Function(
pt + pf_t[(pf_s -
1) *
(_n +
2) +
pa + 1],
pt,
pa,
),
0,
Range(
pf_c,
_n + 1 - pf_s,
1,
),
),
),
pf_row,
pf_c,
),
Map(Function(0, pf_z), Range(0, _n + 2 - 1, 1)),
Range(0, _n + 2 - 1, 1),
),
), pf_t, pf_s),
Fold(
Function(ReplaceAt(pf_row, pf_c + 1, 1), pf_row, pf_c),
Map(Function(0, pf_z), Range(0, _n + 2 - 1, 1)),
Range(0, _n + 2 - 1, 1),
),
Range(1, _n + 1 - 1, 1),
)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
Enumeration
About
| Crosswalk | OEIS |
|---|