SignedReflectionLength

The least number of reflections of the hyperoctahedral group with product p: n minus the positive cycles.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001769via SignedPermutation✓↗

SignedReflectionLength(p)The least number of reflections of the hyperoctahedral group with product p: n minus the positive cycles.

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 St001769. The codimension of the fixed space, which is n less the number of positive cycles. Not ReflectionLength, which counts transpositions of an unsigned permutation.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Length(_x) - Count(Filter(
                                                    Range(1, Length(_x)),
                                                    Function(Count(Filter(
                                                                           Range(1, Length(_x)),
                                                                           Function(Abs(Fold(
                                                                                              Function(-_x[-w] if w < 0 else _x[w], w, j),
                                                                                              i,
                                                                                              Range(1, m, 1),
                                                                                            )) <
                                                                                    i, m),
                                                                         )) == 0 &&
                                                             Count(Filter(
                                                                           Range(1, Length(_x)),
                                                                           Function(Fold(
                                                                                          Function(-_x[-w] if w < 0 else _x[w], w, j),
                                                                                          i,
                                                                                          Range(1, m, 1),
                                                                                        ) ==
                                                                                    -i, m),
                                                                         )) == 0, i),
                                                  ))

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001769

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001769

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001769

About

Examples	4
Crosswalk	FindStat