Limit Behavior of Order Statistics on Cycle Lengths of Random A-Permutations

We consider a random permutation τₙ uniformly distributed on the set of all permutations of degree n whose cycle lengths lie in a fixed set A (the so-called A-permutations). Let ζₙ be the total number of cycles, and let ηₙ(1)≤ηₙ(2)≤≤ηₙ(ζₙ) be the ordered sample of cycle lengths of the permutation τₙ. We consider a class of sets A with positive density in the set of natural numbers. We study the asymptotic behavior of ηₙ(m) with numbers m in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence ηₙ(m) dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case A= N.