We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2,.... In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 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 exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).
"Reversible Markov structures on divisible set partitions." J. Appl. Probab. 52 (3) 622 - 635, September 2015. https://doi.org/10.1239/jap/1445543836