## Journal of Applied Probability

### Reversible Markov structures on divisible set partitions

#### 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,.... 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).

#### Article information

Source
J. Appl. Probab. Volume 52, Number 3 (2015), 622-635.

Dates
First available in Project Euclid: 22 October 2015

https://projecteuclid.org/euclid.jap/1445543836

Digital Object Identifier
doi:10.1239/jap/1445543836

Mathematical Reviews number (MathSciNet)
MR3414981

Zentralblatt MATH identifier
1361.60010

#### Citation

Crane, Harry; McCullagh, Peter. Reversible Markov structures on divisible set partitions. J. Appl. Probab. 52 (2015), no. 3, 622--635. doi:10.1239/jap/1445543836. https://projecteuclid.org/euclid.jap/1445543836

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