Random mappings with component sizes from a given set

The paper is concerned with single-valued mappings from the set of $n$ labeled elements into itself such that the sizes of connected components of the graph corresponding to each mapping lie in a given countable set of positive integers. We find the asymptotic behavior for the number of all such mappings as $n\to\nobreak \infty$. As a conjecture, we formulate sufficient conditions for the convergence of the distribution of the number of components in a random equiprobable mapping of the above form to the normal law (in the local setting). We consider particular cases where this conjecture applies and derive corollaries from it. Conditions are given for the convergence of the distribution of the number of components of a given size to a Poisson distribution law.