Cycle structure of random standardized permutations

TL;DR

This paper investigates the cycle structure of random standardized permutations, revealing diverse limit distributions for cycle counts, including Poisson, geometric, and Poisson-Dirichlet, using exact joint distribution formulas and the method of moments.

Contribution

It provides the first exact joint distribution of cycle counts in this model and establishes new limit theorems for small and large cycles.

Findings

Small cycle counts converge to Poisson or geometric distributions.

Large cycles follow the Poisson-Dirichlet distribution.

Total cycle count obeys a central limit theorem.

Abstract

In this article, we study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. First we prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally we prove a central limit theorem for the total number of cycles.