Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation

We consider the set of $n$-permutations with cycle lengths belonging to some fixed set $A$ of natural numbers (so-called $A$-permutations). Let random permutation $\tau_n$ be uniformly distributed on this set. For some class of sets $A$ we find the asymptotics with remainder term for moments of total cycle number of $\tau_n$.