Consider a "random" bipartite directed graph where (1) on each side, the set of vertices has cardinality n and (2) for each vertex i, we add one (and only one) directed edge i->j at random (drawn uniformly over the n possible directed edges).

Clearly, for all n, and any possible realization of the random graph, there is at least one cycle. 

But what's the expected number of cycles when n tends to infinity? Given k (even), what's the expected number of cycles of size k when n tends to infinity?

I suppose this problem is standard... I would be greatful if one could give me references on this.

Thanks!