Chef and cycle cover of a directed graph

Chef once wanted to count the number of vertex disjoint cycle covers of a directed graph. A cycle cover is a set of cycles covering the set of vertices (A set of cycles S is said to be covering set of vertices if each vertex lies in at least one of the cycles in S). A vertex disjoint cycle cover is a cycle cover in which no two cycles have any common vertex. A cycle is a sequence of t > 1 distinct