NonInversions

Pairs i < j with p(i) < p(j).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000246via Permutation✓↗

NonInversions(p)Pairs i < j with p(i) < p(j).

SageMathenumeratio_stat("Permutation", p, "NonInversions")

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) < 2 else Sum(Map(
                                  Function(
                                            Count(Filter(
                                                          Range(
                                                                 i + 1,
                                                                 Length(_x),
                                                               ),
                                                          Function(
                                                                    _x[i] <
                                                                    _x[j],
                                                                    j,
                                                                  ),
                                                        )),
                                            i,
                                          ),
                                  Range(1, Length(_x)),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000246

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000246

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000246

About

Examples	6
Checked against	SageMath 6 agreemathlib4 3 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Pairs i < j with l[i] < l[j], counted over Fin pairs of the list