Limit theorem for the general number of cycles in a random $A$-permutation

Let $S_n$ be the symmetric group of all permutations of degree $n, A$ be some nonempty subset of the set of natural numbers $N$, and let $T_n=T_n(A)$ be the set of all permutations from $S_n$ with cycle lengths from $A$. The permutations from $T_n$ are called $A$-permutations. Let $\zeta_n$ be the general number of cycles in a random permutation uniformly distributed on $T_n$. In this paper, we find the way to prove the limit theorem for $\zeta_n$ starting with the asymptotics of $|T_n|$. The limit theorem obtained here is new in a number of cases when the asymptotics of $|T_n|$ is known but the limit theorem for $\zeta_n$ has not yet been proven by other methods. As has been noted by the author, $|T_n|/n!$ is the Karamata regularly varying function with index $\sigma-1$, where $\sigma>0$ is the density of the set $A$, in a number of papers of different authors. Proof of the limit theorem for $\zeta_n$ is the main goal of this paper, assuming none of the additional restrictions typical of previous investigations.