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

Let Λ^k _n denote the set of (0, 1)-matrices of order n with exactly k ones in each row and column. Let J_i be such that Λⁱ _i = {J_i} and for A ∈ Λ^k _n define Ā ∈ Λ^n-k _n by Ā = J_n - A. We are interested in the matrices in Λ^k _n which maximise the permanent function. Consider the sets Mn = {A ∈ Λ^k _n: per(A) ≥ per(B), for all B ∈ Λ^k _n}, M̄kn = {A ∈ Λ^k _n: per(Ā) ≥ per(B̄), for all B ∈ Λ^k _n}. For k fixed and n sufficiently large we prove the following results. 1. Modulo permutations of the rows and columns, every member of M^k _n ∪ M̄^k _n is a direct sum of matrices of bounded size of which fewer than k differ from J_k. 2. A ∈ M^k _n if and only if A ⊕ J_k ∈ M^k _n+k. 3. A ∈ M̄^k _n if and only if A ⊕ J_k ∈ M̄^k _n+k. 4. M3n = M̄3n if n ≡ 0 or 1 (mod3) but M3n ∩ M̄³ _n = ∅ if n ≡ 2(mod3). We also conjecture the exact composition of M̄^k _n for large n, which is equivalent to identifying regular bipartite graphs with the maximum number of 4-cycles.