Exchangeable Gibbs partitions and Stirling triangles

Abstract

For two collections of nonnegative and suitably normalized weights W = (W_j) and V = (V_n,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {A_j, j ≤ k} the probability V_n,k $V_{n,k} W_{\left| {A_1 } \right|} \cdots W_{\left| {A_k } \right|}$, where |A_j| is the number of elements of A_j. We impose constraints on the weights by assuming that the resulting random partitions Π _n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Π_n as n tends to infinity. Bibliography: 29 titles.