On the distribution of the $m$th maximal cycle lengths of random $A$-permutations

A. L. Yakymiv

Diskretnaya Matematika, Tome 17 (2005) no. 4, pp. 40-58

Résumé

Let $S_n$ be the symmetric group of all permutations of degree $n$, $A$ be some subset of the set of natural numbers $\mathbf N$, and $T_n=T_n(A)$ be the set of all permutations of $S_n$ with cycle lengths belonging to $A$. The permutations of $T_n$ are called $A$-permutations. We consider a wide class of the sets $A$ with the asymptotic density $\sigma>0$. In this article, the limit distributions are obtained for $\mu_{m}(n)/n$ as $n\to\infty$ and $m\in\mathbf N$ is fixed. Here $\mu_{m}(n)$ is the length of the $m$th maximal cycle in a random permutation uniformly distributed on $T_n$. It is shown here that these limit distributions coincide with the limit distributions of the corresponding functionals of the random permutations in the Ewens model with parameter $\sigma$. This research was supported by the Russian Foundation for Basic Research, grant 05–01–00583, and by the Program of the President of the Russian Federation for support of leading scientific schools, grant 1758.2003.1.