CentralizerSize

The size of the centralizer of a permutation with these cycle lengths: the product of the parts times the product of the factorials of the multiplicities.

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CentralizerSize(partition)The size of the centralizer of a permutation with these cycle lengths: the product of the parts times the product of the factorials of the multiplicities.

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.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

1 if Length(_x) < 1 else Product(_x) * Product(Map(
                                                    Function(Count(Filter(
                                                                           Range(1, i),
                                                                           Function(
                                                                                     _x[j] == _x[i],
                                                                                     j,
                                                                                   ),
                                                                         )), i),
                                                    Range(1, Length(_x)),
                                                  ))

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{CentralizerSize}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$18$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000184

FindStat's value for [2]

live ↗

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

Known value · FindStat St000184

FindStat's value for [3]

live ↗

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

Known value · FindStat St000184

About

Examples	5
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