NontrivialCycles

Cycles of size at least two: the cycles less the fixed points.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000162via Permutation✓↗

NontrivialCycles(p)Cycles of size at least two: the cycles less the fixed points.

SageMathenumeratio_stat("Permutation", p, "NontrivialCycles")

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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(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),
                                                 ),
                                         ), Function(c > 1, c)))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000162

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000162

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000162

About

Examples	5
Checked against	SageMath 5 agreemathlib4 2 agree
Crosswalk	FindStatSageMath

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"); Equiv.Perm.cycleType omits the 1-cycles