Reversible Markov structures on divisible set partitions

TL;DR

This paper introduces reversible Markov chains on $k$-divisible set partitions, providing explicit seating rules and a new notion of Markovian partition structures that are consistent under a Markovian deletion process.

Contribution

It develops a framework for exchangeable $k$-divisible partitions with Markovian dynamics, extending previous models to include reversibility and Markovian deletion procedures.

Findings

Explicit Chinese restaurant-type seating rules for $k$-divisible partitions.

Introduction of Markovian partition structures with reversible Markov chains.

Chains are consistent under Markovian deletion and extend previous models.

Abstract

We study $k$-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer $k=1,2,\ldots$. 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 {\em Markovian partition structures}, which are ensembles of exchangeable Markov chains on $k$-divisible partitions that are consistent under a random process of {\em Markovian deletion}. The Markov chains we study are reversible and refine the class of Markov chains introduced in {\em J.\ Appl.\ Probab.}~{\bf48}(3):778--791.