A lower bound on the maximum permanent in kn

Let Pk n be the maximum value achieved by the permanent over k n, the set of (0, 1)-matrices of order n with exactly k ones in each row and column. Brègman proved that Pk n k!n/k . It is shown here that Pk n k!t r! where n = tk + r and 0 r < k. From this simple bound we derive that (P k n ) 1/n ∼ k!1/k whenever k = o(n) and deduce a number of structural results about matrices which achieve Pk n . These include restrictions for large n and k on the number of components which may be drawn from k k+c for a constant c 1. Our results can be directly applied to maximisation problems dealing with the number of extensions to Latin rectangles or the number of perfect matchings in regular bipartite graphs. © 2003 Published by Elsevier Inc.