WeakExcedanceMidPointsOfOccurrencesOf321

Positions i with p(i) >= i that are the middle of a 321 occurrence.

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WeakExcedanceMidPointsOfOccurrencesOf321(p)Positions i with p(i) >= i that are the middle of a 321 occurrence.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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 Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] >= i && _x[i] < Max(Map(
                                                                               Function(_x[j], j),
                                                                               Range(1, i),
                                                                             )) &&
                                                              _x[i] > Min(Map(
                                                                               Function(_x[j], j),
                                                                               Range(i, Length(_x)),
                                                                             )), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{WeakExcedanceMidPointsOfOccurrencesOf321}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{WeakExcedanceMidPointsOfOccurrencesOf321}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{WeakExcedanceMidPointsOfOccurrencesOf321})$Out$0$

Known value · FindStat St000373

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{WeakExcedanceMidPointsOfOccurrencesOf321})$Out$0$

Known value · FindStat St000373

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{WeakExcedanceMidPointsOfOccurrencesOf321})$Out$0$

Known value · FindStat St000373

About

Examples	5
Crosswalk	FindStat