OddInversions

Inversions i < j with j - i odd.

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OddInversions(p)Inversions i < j with j - i odd.

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(
                                            Sum(Map(
                                                     Function(1, j),
                                                     Filter(
                                                             Range(
                                                                    i + 1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       _x[i] >
                                                                       _x[j] && (j -
                                                                            i) %
                                                                           2 == 1,
                                                                       j,
                                                                     ),
                                                           ),
                                                   )),
                                            i,
                                          ),
                                  Range(1, Length(_x) - 1),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000539

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000539

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

live ↗

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

Known value · FindStat St000539

About

Examples	6
Crosswalk	FindStat