InverseDescentParkingFunctionCount

Parking functions of size n whose diagonal reading word has the inverse descent composition c.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000285via IntegerComposition✓↗

InverseDescentParkingFunctionCount(composition)Parking functions of size n whose diagonal reading word has the inverse descent composition c.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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.
• FindStat's St000285 is the size of that preimage; the count here is the equivalent word count, 1/(n+1) of the words on 1 .. n+1 that fall within each block and rise between blocks.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

Apply(
       (cuts: list<integer>) => 1 if Sum(_x) < 2 else Max(Sum(Fold(
                                                                    Function(Map(
                                                                                  Function(Sum(Map(
                                                                                                    Function(a[u], u),
                                                                                                    Filter(
                                                                                                            Range(1, Sum(_x) + 1),
                                                                                                            Function(u <= w if Count(Filter(cuts, Function(c == b, c))) >
                                                                                                                               0 else u > w, u),
                                                                                                          ),
                                                                                                  )), w),
                                                                                  Range(1, Sum(_x) + 1),
                                                                                ), a, b),
                                                                    Map(
                                                                         Function(1, w),
                                                                         Range(
                                                                                1,
                                                                                Sum(_x) + 1,
                                                                              ),
                                                                       ),
                                                                    Range(
                                                                           1,
                                                                           Sum(_x) -
                                                                           1,
                                                                           1,
                                                                         ),
                                                                  )), 0) / (Sum(_x) +
                                                                           1),
       Fold(Function(Join(a, [Last(a) + b]), a, b), [0], _x),
     )

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{InverseDescentParkingFunctionCount}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$55\,825$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{InverseDescentParkingFunctionCount})$Out$1$

Known value · FindStat St000285

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{InverseDescentParkingFunctionCount})$Out$1$

Known value · FindStat St000285

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([4]), \text{InverseDescentParkingFunctionCount})$Out$1$

Known value · FindStat St000285

About

Examples	4
Crosswalk	FindStat