Coherent Random Allocations, and the Ewens–Pitman Formula

Assume that there is a random number $K$ of positive integer random variables $S_1,\dots,S_K$ which are conditionally independent, given $K$, and all have identical distributions. A random integer partition $N=S_1+S_2+\ldots+S_K$ arises, and we denote by $P_N$ the conditional distribution of this partition, for a fixed value of $N$. We prove that the distributions $\{P_N\}_{N=1}^\infty$ form a partition structure in the sense of Kingman if, and only if, they are governed by the Ewens–Pitman Formula. The latter generalizes the celebrated Ewens Sampling Formula which has numerous applications in pure and applied mathematics.
The distributions of random variables $K$ and $S_j$ belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence, every Ewens–Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions.