SkewHookPositionSum

The sum of the skew hook positions: after two up steps and a down step, the down step's rank among the down steps; after two down steps and an up step, the rank of the second down step.

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SkewHookPositionSum(path)The sum of the skew hook positions: after two up steps and a down step, the down step's rank among the down steps; after two down steps and an up step, the rank of the second down step.

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

0 if Length(_x) < 3 else Sum(Map(
                                  Function(Count(Filter(
                                                         Range(1, i),
                                                         Function(_x[d] == 0, d),
                                                       )) if _x[i - 2] == 1 && _x[i - 1] ==
                                                                              1 &&
                                                            _x[i] == 0 else Count(Filter(
                                                                                Range(1, i - 1),
                                                                                Function(_x[d] == 0, d),
                                                                              )) if _x[i - 2] ==
                                                                                   0 &&
                                                                                   _x[i - 1] ==
                                                                                   0 &&
                                                                                   _x[i] == 1 else 0, i),
                                  Range(3, Length(_x)),
                                ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000946

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

live ↗

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

Known value · FindStat St000946

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

live ↗

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

Known value · FindStat St000946

About

Examples	5
Crosswalk	FindStat