Random derangements and the Ewens Sampling Formula

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.