cardinalityOfConjugacyClass(Partition) -- the size of the conjugacy classes of S_n

Synopsis

Function: cardinalityOfConjugacyClass
Usage:

cardinalityOfConjugacyClass p
Inputs:
- p, an instance of the type Partition, a partition that indexes a conjugacy class of S_n
Outputs:
- an integer, the size of the conjugacy class

Description

The formula for this classes is obtained by the Orbit-Stabilizer lemma applied for S_n with the action of conjugation.

For a partition $p$ this formula is $n!/(\prod_i (\lambda_i )!i^\lambda_i$, where $\lambda_i$ denotes the number of parts in $p$ that are equal to $i$.

i1 : p1 = new Partition from {3,2,1}

o1 = Partition{3, 2, 1}

o1 : Partition

i2 : cardinalityOfConjugacyClass p1

o2 = 120

i3 : p2 = new Partition from {1,1,1,1,1}

o3 = Partition{1, 1, 1, 1, 1}

o3 : Partition

i4 : cardinalityOfConjugacyClass p2

o4 = 1