Counting 1-Factors in Regular Bipartite Graphs

We show that anyk-regular bipartite graph with 2nvertices has at least(k−1)k-1kk-2nperfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integern×nmatrix with each row and column sum equal tok. For anyk, the base (k−1)k−1/kk−2is largest possible.