The cyclic structure of random permutations

Abstract

Letα r denote the number of cycles of length r in a random permutation, taking its values with equal probability from among the set Sn of all permutations of length n. In this paper we study the limiting behavior of linear combinations of random permutationsα 1, ...,α r having the form $$\zeta _{n, r} = c_{r1^{a_1 } } + ... + c_{rr} a_r $$ in the case when n, r→∞. We shall show that the class of limit distributions forξ n,r as n, r→∞ and r In r/h→0 coincides with the class of unbounded divisible distributions. For the random variables ηn, r=α 1+2α 2+... rα r, equal to the number of elements in the permutation contained in cycles of length not exceeding r, we find' limit distributions of the form r In r/n→0 and r=γ n, 0<γ<1.