Some results towards the Dittert conjecture on permanents

Let KnKn denote the convex set consisting of all real nonnegative n×nn×n matrices whose entries have sum nn. For A∈KnA∈Kn with row sums r1,…,rnr1,…,rn and column sums c1,…,cnc1,…,cn, define ϕ(A)=∏i=1nri+∏j=1ncj-per(A). Dittert’s conjecture asserts that the maximum of ϕϕ on KnKn occurs uniquely at Jn=[1/n]n×nJn=[1/n]n×n. In this paper, we prove:(i)if A∈KnA∈Kn is partly decomposable then ϕ(A)<ϕ(Jn)ϕ(A)<ϕ(Jn);(ii)if the zeroes in A∈KnA∈Kn form a block then AA is not a ϕϕ-maximising matrix;(iii)ϕ(A)<ϕ(Jn)ϕ(A)<ϕ(Jn) unless δ:=per(Jn)-per(A)⩽O(n4e-2n)δ:=per(Jn)-per(A)⩽O(n4e-2n) andk-∑i∈αri<2δk,k-∑i∈βci<2δkand∑i∈α,j∈βaij