Profiles of Permutations

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we show that the asymptotic expected number of cycles of random permutations of [n] with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all 12 logn+O(1), and the variance is of the same order. Furthermore, we show that in permutations of [n] chosen from the Ewens distribution with parameter σ, the probability of a random element being in a cycle longer than γn approaches (1−γ)σ for large n. The same limit law holds for permutations with cycles carrying multiplicative weights with average σ. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.