AdmissibleInversions

Pairs i < j with p(i) > p(j) and either p(i-1) < p(i) or some entry between them larger than p(i).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000463via Permutation✓↗

AdmissibleInversions(p)Pairs i < j with p(i) > p(j) and either p(i-1) < p(i) or some entry between them larger than p(i).

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.
• FindStat's St000866 is the Shareshian-Wachs version, which differs.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000463

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000463

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000463

About

Examples	6
Crosswalk	FindStat