OccurrencesOfInitial123

Triples 1 < j < k ordered as 123 (|123).

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OccurrencesOfInitial123(p)Triples 1 < j < k ordered as 123 (|123).

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(
                                                                 j + 1,
                                                                 Length(_x),
                                                               ),
                                                          Function(
                                                                    _x[1] <
                                                                    _x[j] && _x[j] <
                                                                        _x[k],
                                                                    k,
                                                                  ),
                                                        )),
                                            j,
                                          ),
                                  Range(2, Length(_x) - 1),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOfInitial123}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOfInitial123}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOfInitial123}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOfInitial123})$Out$0$

Known value · FindStat St000804

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOfInitial123})$Out$0$

Known value · FindStat St000804

FindStat's value for [1,2,3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2, 3]), \text{OccurrencesOfInitial123})$Out$1$

Known value · FindStat St000804

About

Examples	6
Crosswalk	FindStat