Computing the number of h-edge spanning forests in complete bipartite graphs

Let f(m,n,h) be the number of spanning forests with h edges in the complete bipartite graph K-m,K-n. Kirchhoff's Matrix Tree Theorem implies f(m,n,m+n-1) = m(n-1)n(m-1) when m >= 1 and n >= 1, since f(m,n,m+n-1) is the number of spanning trees in K-m,K-n. In this paper, we give an algorithm for computing f(m,n,h) for general m, n, h. We implement this algorithm and use it to compute all non-zero f(m,n,h) when m <= 50 and n <= 50 in under 2 days.