# Ewens distribution on the symmetric group

From Groupprops

## Contents

## 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

- Ewens distribution on the set of unordered integer partitions simply adds us the Ewens measure values on all elements in the conjugacy class corresponding to that partition.

## 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 |