OccurrencesOf213

Triples i < j < k with p(j) < p(i) < p(k).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000218via Permutation✓↗

OccurrencesOf213(p)Triples i < j < k with p(j) < p(i) < p(k).

SageMathenumeratio_stat("Permutation", p, "OccurrencesOf213")

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

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOf213}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf213}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOf213}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$1$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{OccurrencesOf213})$Out$0$

Known value · FindStat St000218

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOf213})$Out$0$

Known value · FindStat St000218

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOf213})$Out$0$

Known value · FindStat St000218

About

Examples	6
Checked against	SageMath 6 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts