Limit theorem concerning random mapping patterns

Mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. Assuming uniform probability distribution on the set of all mapping patterns onn points, we obtain limit distributions of some characteristics associated with the graphs of mapping patterns (connected and disconnected), asn. In particular, we study the number of points belonging to cycles, the number of cycles and components having prescribed (fixed) number of points and the total number of components.