Some results towards the Dittert conjecture on permanents

Let K-n denote the convex set consisting of all real nonnegative n x n matrices whose entries have sum n For A is an element of K-n with row sums r(1)..... r(n) and column sums c(1).....c(n), define phi(A) = Pi(n)(i=1) r(i) + Pi(n)(j=1)c(j) - per(A). Dittert s conjecture asserts that the maximum of phi on K-n occurs uniquely at j(n) = [1/n](nxn). In this paper, we prove: (i) if A is an element of K-n is partly decomposable then phi(A) <phi(J(n)); (ii) if the zeroes in A is an element of K-n form a block then A is not a phi-maximising matrix: (iii) phi(A) <phi(J(n)) unless delta := per upsilon n)-per(A)