PositionsOfLetterThreeIn132

Positions that are the middle of an occurrence of 132.

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PositionsOfLetterThreeIn132(p)Positions that are the middle of an occurrence of 132.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 3 else Count(Filter(
                                       Range(2, Length(_x) - 1),
                                       Function(Sum(Map(
                                                         Function(Count(Filter(
                                                                                Range(j + 1, Length(_x)),
                                                                                Function(_x[i] < _x[k] &&
                                                                                         _x[k] < _x[j], k),
                                                                              )), i),
                                                         Range(1, j - 1),
                                                       )) > 0, j),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{PositionsOfLetterThreeIn132}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $312$, as a plain list

live ↗

In$\mathrm{PositionsOfLetterThreeIn132}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{PositionsOfLetterThreeIn132}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$1$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{PositionsOfLetterThreeIn132})$Out$0$

Known value · FindStat St001683

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{PositionsOfLetterThreeIn132})$Out$0$

Known value · FindStat St001683

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{PositionsOfLetterThreeIn132})$Out$0$

Known value · FindStat St001683

About

Examples	6
Crosswalk	FindStat