Random derangements and the Ewens Sampling Formula

TL;DR

This paper analyzes derangements under the Ewens distribution, deriving moments, distributions, and asymptotic behaviors, and introduces a Markov chain method for efficient simulation of derangements.

Contribution

It provides new analytical results on cycle counts and distributions, and develops a Markov chain approach for simulating derangements efficiently.

Findings

Derived moments and marginal distributions of cycle counts.

Established asymptotic distributions for large n.

Developed a Markov chain for derangement simulation.

Abstract

We study derangements of $\{1,2,\ldots,n\}$ under the Ewens distribution with parameter $\theta$. We give the moments and marginal distributions of the cycle counts, the number of cycles, and asymptotic distributions for large $n$. We develop a $\{0,1\}$-valued non-homogeneous Markov chain with the property that the counts of lengths of spacings between the 1s have the derangement distribution. This chain, an analog of the so-called Feller Coupling, provides a simple way to simulate derangements in time independent of $\theta$ for a given $n$ and linear in the size of the derangement.