ON THE ORDER OF RANDOM PERMUTATION WITH CYCLE WEIGHTS.

Let Ord(τ) be the order of an element in the group Sn of permutations of an n-element set X. The present paper is concerned with the so-called general parametric model of a random permutation; according to this model an arbitrary fixed permutation τ from Sn is observed with the probability where ui is the number of cycles of length i of the permutation τ, {θi, i ∈ N} are some nonnegative parameters (the weights of cycles of length i of the permutation τ), and H(n) is the corresponding normalizing factor. We assume that an arbitrary permutation τn has such a distribution. The function p(n) = H(n)/n is assumed to be RO-varying at infinity with the lower index exceeding -1 (in particular, it can vary regularly), and the sequence {θi, i ε N} is bounded. Under these assumptions it is shown that the random variable lnOrd(τ_n). In particular, this scheme subsumes the class of random A-permutations (i.e., when θi = x{i A}), where A is an arbitrary fixed subset of the positive integers. This scheme also includes the Ewens model of random permutation, where θi ≡ θ > 0 for any i ∈ N. The limit theorem we prove here extends some previous results for these schemes. In particular, with θi ≡ 1 for any i ∈ N, the result just mentioned implies the well-known Erdos-Turan limit theorem. [ABSTRACT FROM AUTHOR]