Random generation under the Ewens distribution

Abstract

The Ewens sampling formula with parameter α is the distribution on Sn which gives each π ∈ Sn weight proportional to α C(π) , where C(π) is the number of cycles of π. We show that, for any fixed α, two Ewens-random permutations generate at least An with high probability. More generally we work out how many permutations are needed for α growing with n. Roughly speaking, two are needed for 0 ≤ α ≪ n 1/2 , three for n 1/2 ≪ α ≪ n 2/3 , etc.