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 t is the following conjugation-invariant measure on the group. For the symmetric group of degree n, it assigns, to any permutation w \in S_n, the value:

\frac{t^{c(w)}}{t(t+1) \dots (t + n - 1)}

where c(w) is the number of cycles in w (here, fixed points are treated as cycles of length 1). The denominator is the Pochhammer symbol (t)_n.

The Ewens distribution differs from the uniform distribution (or counting measure) where all elements of the symmetric group are assigned the value 1/(n!).

Related notions

Particular cases

n = 2

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
() 2 1 t/(t + 1) t/(t + 1)
(1,2) 1 1 1/(t + 1) 1/(t + 1)

n = 3

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
() 3 1 t^2/(t + 1)(t + 2) t^2/(t + 1)(t + 2)
(1,2) 2 3 t/(t + 1)(t + 2) 3t/(t+1)(t+2)
(1,2,3) 1 2 1/(t + 1)(t + 2) 2/(t + 1)(t + 2)

n = 4

Conjugacy class representative Number of cycles Number of elements Measure of single element in conjugacy class Total measure of conjugacy class
() || 4 || 1 || <math>t^3/(t + 1)(t + 2)(t + 3) t^3/(t + 1)(t + 2)(t + 3)
(1,2) 3 6 t^2/(t + 1)(t + 2)(t + 3) 6t^2/(t + 1)(t + 2)(t + 3)
(1,2,3) 2 8 t/(t + 1)(t + 2)(t + 3) 8t/(t + 1)(t + 2)(t + 3)
(1,2)(3,4) 2 3 t/(t + 1)(t + 2)(t + 3) 3t/(t + 1)(t + 2)(t + 3)
(1,2,3,4) 1 6 1/(t + 1)(t + 2)(t + 3) 6/(t + 1)(t + 2)(t + 3)