Maximising the permanent and complementary permanent of (0, 1)-matrices with constant line sum

Let n denote the set of (0; 1)-matrices of order n with exactly k ones in each row and column. Let Ji be such that i = {Ji} and for A∈ n de ne A∈ n−k n by A = Jn − A. We are interested in the matrices in n which maximise the permanent function. Consider the sets M n = {A∈ n: per(A)¿per(B); for all B∈ n}; M k n = {A∈ n: per(A)¿per(B); for all B∈ n}: For k xed and n su ciently large we prove the following results. 1. Modulo permutations of the rows and columns, every member of M n ∪M n is a direct sum of matrices of bounded size of which fewer than k di er from Jk . 2. A∈Mk n if and only if A⊕ Jk ∈Mk n+k . 3. A∈M n if and only if A⊕ Jk ∈M n+k . 4. M 3 n =M 3 n if n ≡ 0 or 1 (mod 3) but M 3 n ∩M 3 n = ∅ if n ≡ 2 (mod 3). We also conjecture the exact composition of M k n for large n, which is equivalent to identifying regular bipartite graphs with the maximum number of 4-cycles. c © 1999 Elsevier Science B.V. All rights reserved.