Markov chains generating random permutations and set partitions

Abstract

The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on $n$ elements proportionally to absorption probabilities ${\theta }^{\left|\pi \right|}$, $\theta >0$, where $\left|\pi \right|$ is the number of cycles of permutation $\pi$. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.