ProperlyAboveDyckPathCount

The Dyck paths of the same semilength weakly above this path, itself excluded.

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ProperlyAboveDyckPathCount(path)The Dyck paths of the same semilength weakly above this path, itself excluded.

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.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

(1 if Length(_x) / 2 < 2 else Apply(
                                     (s: list<integer>) => Max(Sum(Fold(
                                                                         Function(Map(
                                                                                       Function(Sum(Map(
                                                                                                         Function(a[u], u),
                                                                                                         Range(Max([w - 1, 1]), Length(_x) / 2),
                                                                                                       )) if w - 1 >=
                                                                                                            b else 0, w),
                                                                                       Range(1, Length(_x) / 2),
                                                                                     ), a, b),
                                                                         Map(
                                                                              Function(1 if w == 1 else 0, w),
                                                                              Range(1, Length(_x) / 2),
                                                                            ),
                                                                         Map(
                                                                              Function(s[i], i),
                                                                              Range(2, Length(_x) / 2),
                                                                            ),
                                                                       )), 0),
                                     Map(
                                          Function(0 if i < 2 else Sum(Map(
                                                                            Function(2 * _x[k] -
                                                                                     1, k),
                                                                            Range(1, i - 1),
                                                                          )), i),
                                          Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[i] == 1, i),
                                                ),
                                        ),
                                   )) - 1

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000419

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

live ↗

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

Known value · FindStat St000419

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0, 1, 0]), \text{ProperlyAboveDyckPathCount})$Out$4$

Known value · FindStat St000419

About

Examples	5
Crosswalk	FindStat