Fragmenting random permutations

Abstract

Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each $n$ a fragmentation process $(\Pi_{n,k}, 1 \leq k \leq n)$ such that $\Pi_{n,k}$ is distributed like the partition generated by cycles of a uniform random permutation of $\{1,2,\ldots,n\}$ conditioned to have $k$ cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.