| Abstract: | The Ewens sampling formula is a family of probability distributions over the space of cycle types of permutations of n objects, indexed by a real parameter θ. In the case θ = 1, where the distribution reduces to that induced by the uniform distribution on all permutations, the joint distributions of the numbers of cycles of lengths less than b = o(n) is extremely well approximated by a product of Poisson distributions, having mean 1/j for cycle length j: the error is super-exponentially small with nb?1. For θ ≠ 1. the analogous approximation, with means adjusted to θ/j, is good, but with error only linear in n?1b. In this article, it is shown that, by choosing the means of the Poisson distributions more carefully, an error quadratic in n?1b can be achieved, and that essentially nothing better is possible. |
