# Ewens distribution on the symmetric group

## 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 $1/(n!)$.

## 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
$() || 4 || 1 || [itex]t^3/(t + 1)(t + 2)(t + 3)$ $t^3/(t + 1)(t + 2)(t + 3)$
$(1,2)$ 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 $1/(t + 1)(t + 2)(t + 3)$ $6/(t + 1)(t + 2)(t + 3)$