TypeBInversionCount

Pairs i < j with p(i) > p(j), plus pairs i <= j with -p(i) > p(j).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001428via SignedPermutation✓↗

TypeBInversionCount(p)Pairs i < j with p(i) > p(j), plus pairs i <= j with -p(i) > p(j).

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.
• The Coxeter length in type B (FindStat St001428).

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Max(Sum(Map(Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       j > i &&
                                                                       _x[i] >
                                                                       _x[j],
                                                                       j,
                                                                     ),
                                                           )),
                                               i,
                                             ), Range(1, Length(_x)))), 0) +
                         Max(Sum(Map(Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(j >= i &&
                                                                      -_x[i] >
                                                                      _x[j], j),
                                                           )),
                                               i,
                                             ), Range(1, Length(_x)))), 0)

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001428

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001428

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001428

About

Examples	4
Crosswalk	FindStat