SignedPermutationOrder

The least m >= 1 with p^m the identity, p acting on -n..n.

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SignedPermutationOrder(p)The least m >= 1 with p^m the identity, p acting on -n..n.

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 St001555. The lcm of the signed orbit lengths of 1..n: a cycle is counted twice when it is negative. Not PermutationOrder, which reads the word as an unsigned permutation.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

1 if Length(_x) < 1 else Fold(
                               Function(LCM(a, b), a, b),
                               1,
                               Map(Function(Min(Filter(
                                                        Range(
                                                               1,
                                                               2 * Length(_x),
                                                               1,
                                                             ),
                                                        Function(Fold(
                                                                       Function(
                                                                                 -_x[-w] if w < 0 else _x[w],
                                                                                 w,
                                                                                 j,
                                                                               ),
                                                                       i,
                                                                       Range(
                                                                              1,
                                                                              m,
                                                                              1,
                                                                            ),
                                                                     ) == i, m),
                                                      )), i), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )),
                             )

Examples

Basic #

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

live ↗

In$\mathrm{SignedPermutationOrder}({\operatorname{SignedPermutation}}([2, -1, -3]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001555

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001555

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001555

About

Examples	4
Crosswalk	FindStat