On the sets of images of $k$-fold iteration of uniform random mapping

The properties of the graph of $k$-fold iteration of uniform random mapping $f\colon \{1,\ldots,n\}\to \{1,\ldots,n\}$ are studied. Some recurrence formulas for the probabilities for a vertex to belong to the set of images $f^k(\{1,\ldots,n\})$ and to the set of the initial vertices in the graph of $f^k$ are obtained.