REFINED APPROXIMATIONS FOR THE EWENS SAMPLING FORMULA

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-theta. In the case theta = 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 theta not-equal 1, the analogous approximation, with means adjusted to theta/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.