SymmetryClassSize

The permutations reachable from p by inverse, reverse and complement: 8 over the symmetries that fix p.

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SymmetryClassSize(p)The permutations reachable from p by inverse, reverse and complement: 8 over the symmetries that fix p.

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

1 if Length(_x) < 1 else 8 / (1 + (1 if Count(Filter(
                                                      Range(1, Length(_x)),
                                                      Function(
                                                                _x[i] !=
                                                                _x[Length(_x) +
                                                                   1 - i],
                                                                i,
                                                              ),
                                                    )) == 0 else 0) +
                              (1 if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[i] != Length(_x) +
                                                                    1 - _x[i], i),
                                                )) == 0 else 0) +
                              (1 if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(
                                                            _x[Length(_x) + 1 -
                                                               i] != Length(_x) +
                                                                    1 - _x[i],
                                                            i,
                                                          ),
                                                )) == 0 else 0) +
                              (1 if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[_x[i]] != i, i),
                                                )) == 0 else 0) +
                              (1 if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[_x[i]] != Length(_x) +
                                                                        1 - i, i),
                                                )) == 0 else 0) +
                              (1 if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[Length(_x) + 1 -
                                                              _x[i]] != i, i),
                                                )) == 0 else 0) + (1 if Count(Filter(
                                                                                     Range(1, Length(_x)),
                                                                                     Function(_x[Length(_x) + 1 -
                                                                                                 _x[i]] !=
                                                                                              Length(_x) + 1 -
                                                                                              i, i),
                                                                                   )) ==
                                                                       0 else 0))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000401

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000401

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000401

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Examples	5
Crosswalk	FindStat