I am trying to count the number of $(d_v,d_c)$ regular bipartite graphs. To be specific, let $n,m,d_v,d_c$ be positive integers such that $$n\times d_v=m\times d_c.$$ Then, what is the number of bipartite graphs $\mathcal{G}=(L\cup R, E)$, where $L$ is the set of left vertices, $R$ is the set of right vertices with the property that each left vertex has $d_v$ edges incident on it, and each right vertex has $d_c$ edges incident on it (allowing parallel edges, i.e., there can be more than one edge for a given pair of vertices)? Can one find a general expression for this?

This problem can be reformulated as a ball and bin problem in the following way: We have a total of $nd_v$ balls, of $n$ colors, with $d_v$ balls of each color. What is the number of ways of putting these balls in $m$ nonidentical bins so that each bin has exactly $d_c$ balls? Let me denote this number $\psi(n,d_v,d_c)$ A quick thought about this gives me the following bounds: $$ \frac{(nd_v)!}{\left(d_v! \right)^n(d_c!)^m}\leq\psi(n,d_v,d_c)\leq \frac{(nd_v)!}{\left(d_v! \right)^n}. $$ Can one do better?

I think there must be some work that has addressed this problem but I have not been able to find anything. Any help would be appreciated.