Cycle type of random permutations: A toolkit

TL;DR

This paper develops a comprehensive toolkit for analyzing the cycle structure of random permutations, establishing new probabilistic results and techniques inspired by number theory sieves.

Contribution

It introduces new methods for understanding cycle distributions in permutations, extending classical results with sieve-inspired techniques.

Findings

Number of cycles of size k is approximately Poisson with mean 1/k

Distribution of cycles in fixed sets I is characterized precisely

Provides a unified framework for cycle analysis in permutations

Abstract

We prove a number of results, new and old, about the cycle type of a random permutation on S_n. Underlying our analysis is the idea that the number of cycles of size k is roughly Poisson distributed with parameter 1/k. In particular, we establish strong results about the distribution of the number of cycles whose lengths lie in a fixed but arbitrary set I. Our techniques are motivated by the theory of sieves in number theory.