PermutationOrder

The order of p in the symmetric group — the lcm of its cycle lengths.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000058via Permutation✓↗Wolfram LanguagePermutationOrder✓↗

PermutationOrder(p)The order of p in the symmetric group — the lcm of its cycle lengths.

SageMathenumeratio_stat("Permutation", p, "PermutationOrder")

Typed signatures

Type	Package	Applies
(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 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.
• Wolfram's PermutationOrder. The bare Order is Wolfram's canonical-ordering comparison Order[a, b], so this head replaces it. The old name is recorded in formerly but does not resolve: Order(p) is now Wolfram's Order.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000058

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000058

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000058

About

Examples	5
Checked against	Wolfram Language 2 agreeSageMath 5 agreemathlib4 2 agree
Crosswalk	FindStatWolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

A 1-based one-line permutation list, rebuilt as an Equiv.Perm (Fin n) (a list that is not a permutation prints "not a permutation"); the order is the lcm of the cycle type (Equiv.Perm.lcm_cycleType), since orderOf is noncomputable