The Cycle Structure of Permutations Without Long Cycles

Abstract

We consider the cycle structure of a random permutation $\sigma$ chosen uniformly from the symmetric group, subject to the constraint that $\sigma$ does not contain cycles of length exceeding $r.$ We prove that under suitable conditions the distribution of the cycle counts is approximately Poisson and obtain an upper bound on the total variation distance between the distributions using Stein's method of exchangeable pairs. Our results extend the recent work of Betz, Sch\"{a}fer, and Zeindler.