A limit theorem for the logarithm of the order of a~random $A$-permutation

A. L. Yakymiv

Diskretnaya Matematika, Tome 22 (2010) no. 1, pp. 126-149

Résumé

In this article, a random permutation $\tau_n$ is considered which is 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). It is assumed that the set $A$ has an asymptotic density $\sigma>0$, and $|k\colon k\leq n,\ k\in A,\ m-k\in A|/n\to\sigma^2$ as $n\to\infty$ uniformly in $m\in[n,Cn]$ for an arbitrary constant $C>1$. The minimum degree of a permutation such that it becomes equal to the identity permutation is called the order of permutation. Let $Z_n$ be the order of a random permutation $\tau_n$. In this article, it is shown that the random variable $\ln Z_n$ is asymptotically normal with mean $l(n)=\sum_{k\in A(n)}\ln(k)/k$ and variance $\sigma\ln^3(n)/3$, where $A(n)=\{k\colon k\in A,\ k\leq n\}$. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations $S_n$ is considered, i.e., where $A$ is equal to the set of positive integers $\mathbb N$.