# 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  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 .

## 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