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Fragmenting random permutations. (English) Zbl 1189.60022
Summary: Problem 1.5.7 from Pitman’s Saint-Flour lecture notes [Combinatorial stochastic processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII – 2002. Lect. Notes in Math. Berlin: Springer (2006; Zbl 1103.60004)]: 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 \({1,2,\dots ,n}\) conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.

60C05 Combinatorial probability
05A17 Combinatorial aspects of partitions of integers
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