StandardizedBiAlternatingInversionNumber

Half of floor(n/2)^2 plus the signed inversions, each pair i < j weighted by (-1)^(i+j).

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StandardizedBiAlternatingInversionNumber(p)Half of floor(n/2)^2 plus the signed inversions, each pair i < j weighted by (-1)^(i+j).

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.
• Half of the sum of (-1)^(i+j) sgn(p(j) - p(i)) over i < j, shifted by floor(n/2)^2.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else (Length(_x) - Length(_x) % 2) / 2 * ((Length(_x) - Length(_x) %
                                                                            2) /
                                                              2 + 1) / 2 + (Sum(Map(
                                                                                    Function(Sum(Map(
                                                                                                      Function(1, j),
                                                                                                      Filter(
                                                                                                              Range(i + 1, Length(_x)),
                                                                                                              Function(_x[i] < _x[j] &&
                                                                                                                       (j - i) %
                                                                                                                       2 ==
                                                                                                                       0, j),
                                                                                                            ),
                                                                                                    )), i),
                                                                                    Range(1, Length(_x) - 1),
                                                                                  )) -
                                                                           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$\mathrm{StandardizedBiAlternatingInversionNumber}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

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

the one-line word $2413$

live ↗

In$\mathrm{StandardizedBiAlternatingInversionNumber}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000677

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000677

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

live ↗

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

Known value · FindStat St000677

About

Examples	6
Crosswalk	FindStat