On some classes of permutations with number-theoretic restrictions on the lengths of cycles

A. I. Pavlov

Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 263-275

Résumé

The set $S_n(M)$ of the permutations of degree $n$ having only cycles with lengths in a fixed set $M$ is investigated. The set $M$ is distinguished in the set of all positive integers by imposing certain number-theoretic conditions. The following assertions are proved. 1) If $|S_n(M)|$ is the cardinality of the finite set $S_n(M)$, then there exist positive constants $A$ and $\gamma$ with $0\gamma1$ such that $\frac{|S_n(M)|}{n!}=An^{\gamma-1}(1+O((\ln n)^{-1/2}(\ln\ln n)^2))$, $n\to\infty$. 2) If the uniform probability distribution is introduced on the finite set $S_n(M)$ and if $\eta_n$ is the number of cycles in a random permutation in $S_n(M)$, then the random variable $\eta_n'=(\eta_n-\gamma\ln n)(\gamma\ln n)^{-1/2}$ is asymptotically normal with parameters 0 and 1 as $n\to\infty$. Bibliography: 4 titles.