A refinement of the Ewens sampling formula

We consider an infinitely-many neutral allelic model of population genetics where all alleles are divided into a finite number of classes, and each class is characterized by its own mutation rate. For this model the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and the problem is to describe the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula in this paper. We discuss a Poisson approximation for the refined Ewens sampling formula and present its derivation by several methods. As an application, we obtain limit theorems for the numbers of alleles in different asymptotic regimes.