On disposal of particles in cells and random mappings of a finite set

We consider the uniform distribution on the set $\mathfrak M_n$ of all mappings of the finite set $\{1,2,\dots,n\}$ into itself. A random mapping from the set $\mathfrak M_n$ has a random number of components $\varkappa_n=\alpha_1+\dots+\alpha_n$, where $\alpha_r$ is the number of components of size $r$. We arrange the components according to their sizes and denote by $S_m$ the size of the $m$th component in the sequence. We prove a normal local limit theorem for $\varkappa_n$, a Poisson limit theorem for $\alpha_r$ and limit theorems for extreme and middle terms of the sequence $S_1,\dots,S_{\varkappa_n}$.