Asymptotics of the Moments of the Number of Cycles of a Random $A$-Permutation

We consider random permutations uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths belong to a fixed set $A$ (the so-called $A$-permutations). In the present paper, we establish an asymptotics of the moments of the total number of cycles and of the number of cycles of given length of this random permutation as $n\to\infty$.