InverseByConjugationCount

The permutations that a fixed permutation of this cycle type conjugates to their inverse: the square roots of its square, a product over the cycle lengths d of the roots of the cycles of length d.

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InverseByConjugationCount(partition)The permutations that a fixed permutation of this cycle type conjugates to their inverse: the square roots of its square, a product over the cycle lengths d of the roots of the cycles of length d.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition 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.
• alpha pi alpha^-1 = pi^-1 exactly when (pi alpha)^2 = alpha^2.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

1 if Length(_x) < 1 else Apply(
                                (m: list<integer>) => Product(Map(Function(Sum(Map(
                                                                                    Function((1 if m[d] == 0 else Product(Range(1, m[d]))) /
                                                                                             ((1 if m[d] - 2s ==
                                                                                                    0 else Product(Range(1, m[d] - 2s))) *
                                                                                              (1 if s == 0 else Product(Range(1, s))) *
                                                                                              2 ^ s) *
                                                                                             d ^ s if d % 2 ==
                                                                                                 1 ||
                                                                                                 m[d] == 2s else 0, s),
                                                                                    Range(0, (m[d] -
                                                                                              m[d] % 2) /
                                                                                             2, 1),
                                                                                  )), d), Range(1, Max(_x)))),
                                Map(Function((Count(Filter(
                                                            Range(1, Length(_x)),
                                                            Function(
                                                                      _x[i] == d,
                                                                      i,
                                                                    ),
                                                          )) if d % 2 == 1 else 0) +
                                             2 * Count(Filter(
                                                               Range(
                                                                      1,
                                                                      Length(_x),
                                                                    ),
                                                               Function(_x[i] ==
                                                                        2d, i),
                                                             )), d), Range(1, Max(_x))),
                              )

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{InverseByConjugationCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$8$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{InverseByConjugationCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001710

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{InverseByConjugationCount})$Out$2$

Known value · FindStat St001710

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{InverseByConjugationCount})$Out$1$

Known value · FindStat St001710

About

Examples	5
Crosswalk	FindStat