OccurrencesOf21Dash3

Triples i, i+1, k with k > i+1 ordered as 213 (21-3).

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OccurrencesOf21Dash3(p)Triples i, i+1, k with k > i+1 ordered as 213 (21-3).

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 Sum(Map(
                                  Function(Count(Filter(
                                                         Range(
                                                                i + 2,
                                                                Length(_x),
                                                              ),
                                                         Function(_x[i + 1] <
                                                                  _x[i] && _x[i] <
                                                                      _x[k], k),
                                                       )), i),
                                  Range(1, Length(_x) - 2),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOf21Dash3}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf21Dash3}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOf21Dash3}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{OccurrencesOf21Dash3})$Out$0$

Known value · FindStat St000355

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOf21Dash3})$Out$0$

Known value · FindStat St000355

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOf21Dash3})$Out$0$

Known value · FindStat St000355

About

Examples	6
Crosswalk	FindStat