Counting 1-factors in regular bipartite graphs

We show that any k-regular bipartite graph with 2n vertices has at least ((k - 1)(k-1)/k(k-2))(n) perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n x n matrix with each row and column sum equal to k. For any k, the base (k - 1)(k-1)/k(k-2) is largest possible. (C) 1998 Academic Press.