It is known that counting perfect matchings in bipartite graphs is $\#P$-hard.
Given a complete bipartite graph $G(U \cup V, E)$, where $|U|=|V|=n$ and a perfect matching $M \subset E$, what is the number of perfect matchings $N$ such that $M \cap N= \phi$?
I want to count such restricted matching as a function of $n$.
Is there a known formula? If not, what is the best asymptotic lower-bound on the number of restricted matchings?