Ewens sampling formulae with and without selection

We shall first consider the random Dirichlet partitioning of the interval into n   fragments at temperature θ>0.θ>0. Using calculus for Dirichlet integrals, pre-asymptotic versions of the Ewens sampling formulae from finite Dirichlet partitions follow up. From these preliminaries, straightforward proofs of the usual sampling formulae from random proportions with Poisson–Dirichlet (PD)(γ)(PD)(γ) distribution can be obtained, while considering the Kingman limit n↗∞n↗∞, θ↘0θ↘0, with nθ=γ>0nθ=γ>0.In this manuscript, the Gibbs version of the Dirichlet partition with symmetric selection is considered. By use of similar series expansion calculus for Dirichlet integrals, closed-form expressions of Ewens sampling formulae in the presence of selection are obtained; special types of Bell polynomials are shown to be involved.