Distribution of the length of aperiodicity segment in the graph of $k$-fold iteration of uniform random mapping

The distribution of the length of the aperiodicity segment in a graph of $k$-fold iteration of random uniform mapping of a finite set is studied. Exact formulas for this distribution are obtained, the limit distribution of the normed length of aperiodicity segment is found for the case when the cardinality of the set tends to infinity.