Fragmenting random permutations

TL;DR

This paper proves the existence of a fragmentation process for uniform random permutations' cycle partitions and extends some results to exchangeable Gibbs partitions.

Contribution

It establishes the existence of a fragmentation process for cycle partitions of uniform permutations and extends findings to exchangeable Gibbs partitions.

Findings

Existence of a fragmentation process for cycle partitions confirmed.

Partial extension of results to exchangeable Gibbs partitions.

Provides a new perspective on partition fragmentation in combinatorics.

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) taking values in the space of partitions of {1,2,...,n} such that \Pi_{n,k} is distributed like the partition generated by cycles of a uniform random permutation of {1,2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.