Random derangements and the Ewens Sampling Formula

We study derangements of 1,2,ldots,n under the Ewens distribution with parameter θ. 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 θ for a given n and linear in the size of the derangement.