Random mappings of finite sets with a known number of components

We consider the class of all one-to-one mappings of an $n$-element set into itself, each of which has exactly $N$ connected components. Letting $n,N\to\infty$, we find that the asymptotic behavior of the mean and variance of the random variable is equal to the number of components of a given size in a mapping that is selected at random and is equiprobable among the elements of the mentioned class, and we prove the Poisson and local normal limit theorems for this random variable. Asymptotic estimates are found for the number of mappings with $N$ components, among which there are exactly $k$ components of a fixed size.