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 $$1/(n!)$$.

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.