InversionSum

The sum of j - i over the inversions i < j.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000347via BinaryWord✓↗St000055via Permutation✓↗

InversionSum(p)The sum of j - i over the inversions i < j.

Typed signatures

Type	Package	Applies
(binary_word | 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(j - i, j),
                                                    Filter(
                                                            Range(
                                                                   i + 1,
                                                                   Length(_x),
                                                                 ),
                                                            Function(
                                                                      _x[i] >
                                                                      _x[j] && True,
                                                                      j,
                                                                    ),
                                                          ),
                                                  )), i),
                                  Range(1, Length(_x) - 1),
                                ))

on BinaryWord, over _x

0 if Length(_x) < 2 else Max(Sum(Map(
                                      Function(Max(Sum(Map(
                                                            Function((j - i) *
                                                                     (1 if _x[i] >
                                                                           _x[j] else 0), j),
                                                            Range(
                                                                   i + 1,
                                                                   Length(_x),
                                                                   1,
                                                                 ),
                                                          )), 0), i),
                                      Range(1, Length(_x) - 1),
                                    )), 0)

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

In$\mathrm{InversionSum}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$3$

the one-line word $2413$

live ↗

In$\mathrm{InversionSum}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$5$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000055

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000055

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000055

About

Examples	6
Crosswalk	FindStat ×2