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

Theory of Probability &Its Applications, Volume 69, Issue 1, Page 117-126, May 2024.
We consider a random permutation $\tau_n$ 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 $\zeta_n$ be the total number of cycles, and let $\eta_n(1)\leq\eta_n(2)\leq\dots\leq\eta_n(\zeta_n)$ be the ordered sample of cycle lengths of the permutation $\tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $\eta_n(m)$ 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 $\eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case $A=\mathbf N$.