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 S_n 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 η_n, 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.