A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can construct a bipartite $G_k$. For bipartite graphs this result implies that the number of $k$-matching has a polynomial-time approximation algorithm. The above results are extended to permanents and hafnians of corresponding matrices.