MinimalInversionsEitherDirection

The smaller of the number of inversions and the number of non-inversions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001511via Permutation✓↗

MinimalInversionsEitherDirection(p)The smaller of the number of inversions and the number of non-inversions.

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.
• Inversions of p against those of its reverse.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001511

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001511

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001511

About

Examples	6
Crosswalk	FindStat