DyckPathsAboveBouncePathCount

Dyck paths of semilength n that stay weakly above the bounce path of the composition: their area sequences are at least 0 .. c_1 - 1, 0 .. c_2 - 1, ....

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DyckPathsAboveBouncePathCount(composition)Dyck paths of semilength n that stay weakly above the bounce path of the composition: their area sequences are at least 0 .. c_1 - 1, 0 .. c_2 - 1, ....

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 titles it the parabolic noncrossing partitions indexed by the composition (Mühle and Williams) and also the size of a nu-Tamari lattice (Bergeron, Ceballos and Pilaud); its code counts these Dyck paths.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

1 if Length(_x) < 1 else Apply((cuts: list<integer>) => Apply(
                                                               (bounceAreas: list<integer>) => 1 if Sum(_x) < 2 else Max(Sum(Fold(
                                                                                                                                   Function(Map(Function(Sum(Map(
                                                                                                                                                                  Function(a[g], g),
                                                                                                                                                                  Range(Max(h - 1, 1), Sum(_x), 1),
                                                                                                                                                                )) if h - 1 >=
                                                                                                                                                                     bounceAreas[b] else 0, h), Range(1, Sum(_x))), a, b),
                                                                                                                                   Map(Function(1 if h == 1 else 0, h), Range(1, Sum(_x))),
                                                                                                                                   Range(2, Sum(_x), 1),
                                                                                                                                 )), 0),
                                                               Map(
                                                                    Function(i -
                                                                             1 -
                                                                             Max(Filter(
                                                                                         cuts,
                                                                                         Function(u < i, u),
                                                                                       )), i),
                                                                    Range(
                                                                           1,
                                                                           Sum(_x),
                                                                         ),
                                                                  ),
                                                             ), Fold(Function(
                                                                              Join(a, [Last(a) + b]),
                                                                              a,
                                                                              b,
                                                                            ), [0], _x))

Examples

Basic #

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

live ↗

In$\mathrm{DyckPathsAboveBouncePathCount}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$609$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001312

FindStat's value for [2]

live ↗

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

Known value · FindStat St001312

FindStat's value for [3]

live ↗

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

Known value · FindStat St001312

About

Examples	4
Crosswalk	FindStat