Random $A$-permutations in a parametric model

We consider permutations of the set $X_n=\{1,2,\ldots,n\}$ such that lengths of all their cycles are elements of the subset $A\subset \{1,2,\ldots\}$. For a parametric probability measure on the set $S_n(A)$ properties of the cycle structure of a random permutation are studied for several types of subset $A$. The problems of testing corresponding statistical hypotheses on the model are analyzed.