On the cycle structure of the product of random maximal cycles

TL;DR

This paper investigates the cycle structure of a permutation formed by multiplying several large random cycles, using Fourier analysis to understand cycle counts and element distribution.

Contribution

It introduces a character-based Fourier transform approach to analyze the cycle structure of products of maximal cycles in permutations.

Findings

Derived the distribution of the number of cycles in the product permutation

Analyzed the distribution of elements from a subset among the cycles

Provided new insights into the structure of permutations formed by maximal cycles

Abstract

The subject of this paper is the cycle structure of the random permutation $\sigma$ of $[N]$, which is the product of $k$ independent random cycles of maximal length $N$. We use the character-based Fourier transform to study the number of cycles of $\sigma$ and also the distribution of the elements of the subset $[\ell]$ among the cycles of $\sigma$.