Ewens distribution on the symmetric group

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Definition

The Ewens measure' or Ewens distribution on the symmetric group (more specifically, the symmetric group on a finite set) with parameter is the following conjugation-invariant measure on the group. For the symmetric group of degree , it assigns, to any permutation , the value:

where is the number of cycles in (here, fixed points are treated as cycles of length ). The denominator is the Pochhammer symbol .

The Ewens distribution differs from the uniform distribution (or counting measure) where all elements of the symmetric group are assigned the value .

Related notions

Particular cases

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
2 1
1 1

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
3 1
2 3
1 2

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
3 6
2 8
2 3
1 6