ParkingFunctionCount

The parking functions supported by the path: semilength factorial over the product of the factorials of the rise lengths.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000014via DyckPath✓↗

ParkingFunctionCount(path)The parking functions supported by the path: semilength factorial over the product of the factorials of the rise lengths.

Typed signatures

Type	Package	Applies
(dyck_path) -> number	combinatorics	always

Domain: Dyck path statistics

Details

• Defined over DyckPath as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a DyckPath and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• A parking function is a path with the cars of each column labeled; the labels must increase up each rise, which leaves n!/(r_1! r_2! ...) of them.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) < 2 else (1 if Length(_x) / 2 == 0 else Product(Range(
                                                                       1,
                                                                       Length(_x) /
                                                                       2,
                                                                     ))) /
                         Product(Map(
                                      Function(
                                                1 if r == 0 else Product(Range(1, r)),
                                                r,
                                              ),
                                      Apply(
                                             (u: list<integer>) => Map(
                                                                        Function(
                                                                                  u[1] if k == 1 else u[k] -
                                                                                                      u[k - 1],
                                                                                  k,
                                                                                ),
                                                                        Range(1, Length(u)),
                                                                      ),
                                             Map(
                                                  Function(Count(Filter(
                                                                         Range(1, i),
                                                                         Function(
                                                                                   _x[k] == 1,
                                                                                   k,
                                                                                 ),
                                                                       )), i),
                                                  Filter(
                                                          Range(1, Length(_x) -
                                                                   1),
                                                          Function(_x[i] == 1 &&
                                                                   _x[i + 1] ==
                                                                   0, i),
                                                        ),
                                                ),
                                           ),
                                    ))

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{ParkingFunctionCount}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$3$

the step word $UDUUDD$

live ↗

In$\mathrm{ParkingFunctionCount}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$3$

Properties #

FindStat's value for [1,0]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0]), \text{ParkingFunctionCount})$Out$1$

Known value · FindStat St000014

FindStat's value for [1,0,1,0]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0]), \text{ParkingFunctionCount})$Out$2$

Known value · FindStat St000014

FindStat's value for [1,1,0,0]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 1, 0, 0]), \text{ParkingFunctionCount})$Out$1$

Known value · FindStat St000014

About

Examples	5
Crosswalk	FindStat