Limit theorems for allocation of particles over different cells with restrictions to the size of the cells

Equiprobable allocation schemes of $n$ indistinguishable or distinguishable particles over $N$ distinguishable cells are considered provided the fillings of the cells take on values in a fixed subset $A$ of the set of nonnegative integers. Local normal and Poisson theorems are proved for the distributions of the number of cells, each of which contains exactly $r$ particles, and for the number of cycles of length $r\in A$ in a permutation selected at random and equiprobable from the set of all permutations of order $n$ with $N$ cycles $(N\le n)$ whose lengths are elements of a set $A\subsetN$. It is assumed that $n,N\to\infty$ in the central domain.