Random permutations with cycle lengths in a given finite set

We consider the class of all permutations of degree $n$ whose cycle lengths are elements of a fixed finite set $A\subset\mathbf N$ such that $\operatorname{card}A\ge2$ and $\operatorname{gcd}\{k\mid k\in A\}=1$. Under the assumption that the permutation $X$ is equiprobably chosen from this class, we obtain a multidimensional local normal theorem for the joint distribution of the numbers of cycles of given sizes in this permutation.
The obtained results are utilised and sharpened in the case where $X$ is an equiprobably chosen solution of the equation $X^r=e$, where $e$ is an identity permutation of degree $n$, $r\ge2$ is a fixed positive integer.