EXCHANGEABLE AND PARTIALLY EXCHANGEABLE RANDOM PARTITIONS

Call a random partition of the positive integers partially exchangeable if for each finite sequence of positive integers n(1),...,n(k), the probability that the partition breaks the first n(1)+...+n(k) integers into k particular classes, of sizes n(1),...,n(k) in order of their first elements, has the same value p(n(1),...,n(k)) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric function p(n(1),... n(k)). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure.