Fragmenting random permutations

Problem 1.5.7 from Pitman's Saint-Flour lecture notes [Π]: Does there exist for each n a fragmentation process (Πn, k, 1 ≥ k ≥ n) such that Πn, k is distributed like the partition generated by cycles of a uniform random permutation of (1,2,…, n) ng conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions. © 2008 Applied Probability Trust.