SignedSortingIndex

The sum, over the signed transpositions that sort p by moving n, then n - 1, and so on into place, of j - i - 1 when i < 0 and j - i otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001821via SignedPermutation✓↗

SignedSortingIndex(p)The sum, over the signed transpositions that sort p by moving n, then n - 1, and so on into place, of j - i - 1 when i < 0 and j - i otherwise.

Typed signatures

Type	Package	Applies
(signed_permutation) -> number	combinatorics	always

Domain: Signed permutation statistics

Details

• Defined over SignedPermutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SignedPermutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• FindStat's St001821. The fold carries the total and the word, as SortingIndex does.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Fold(Function(Apply(
                                              Function((
                                                         a[1] +
                                                         (b + t - 1 if a[2][t] <
                                                                       0 else b -
                                                                             t),
                                                         ReplaceAt(ReplaceAt(
                                                                              a[2],
                                                                              t,
                                                                              -a[2][b] if a[2][t] < 0 else a[2][b],
                                                                            ), b, b),
                                                       ), t),
                                              Sum(Map(
                                                       Function(j if Abs(a[2][j]) ==
                                                                     b else 0, j),
                                                       Range(1, Length(_x)),
                                                     )),
                                            ), a, b), (0, _x), Range(
                                                                     Length(_x),
                                                                     1,
                                                                     -1,
                                                                   ))[1]

Examples

Basic #

the signed word $2\,\bar{1}\,\bar{3}$

live ↗

In$\mathrm{SignedSortingIndex}({\operatorname{SignedPermutation}}([2, -1, -3]))$Out$7$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001821

FindStat's value for [-1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([-1]), \text{SignedSortingIndex})$Out$1$

Known value · FindStat St001821

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001821

About

Examples	4
Crosswalk	FindStat