Reversible Markov structures on divisible set partitions

Abstract

We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork> 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeablek-divisible partitions that are consistent under random deletion. We further introduce the notion ofMarkovian partition structures, which are ensembles of exchangeable Markov chains onk-divisible partitions that are consistent under a random process ofMarkovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).