On the limit distribution of the number of cycles and the logarithm of the order of a~class of permutations

A. I. Pavlov

Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 539-567

Résumé

Let $S_n$ be the symmetric group of degree $n$ and let $S_n^{(k)}$ be the set of permutations $a\in S_n$ such that the equation $x^k=a$ has a solution $x\in S_n$. Consider the uniform probability distribution on the set $S_n^{(k)}$. This article investigates the limit distributions on $S_n^{(k)}$, as $n\to\infty$ and for fixed $k\geqslant2$, of the random variables $\xi_s$, $\eta$, and $\zeta$, where $\xi_s$ is the number of cycles of length $s$, $\eta$ is the number of all cycles, and $\zeta$ is the logarithm of the order of a random permutation $a\in S_n^{(k)}$. Bibliography: 5 titles.