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 . 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 ) distribution can be obtained, while considering the Kingman limit n ↗ ∞ , θ ↘ 0 , with n θ = γ > 0 . s 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.