The Cycle Structure of Permutations Without Long Cycles

TL;DR

This paper studies the cycle structure of permutations without long cycles, showing that their cycle counts are approximately Poisson distributed and providing bounds on their distributional distance.

Contribution

It extends previous work by establishing Poisson approximation for cycle counts in constrained permutations using Stein's method.

Findings

Cycle counts are approximately Poisson distributed.

Provides bounds on total variation distance.

Extends prior results to new permutation constraints.

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.