Random derangements and the Ewens Sampling Formula
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We study derangements of {1,2,...,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.
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