Some results towards the Dittert conjecture on permanents
Let Kn denote the convex set consisting of all real nonnegative
n×n matrices whose entries have sum n. For A in Kn with
row sums r1,...,rn and column sums c1,...,cn, define
φ(A)=Πi=1n ri +Πj=1n cj -per(A). Dittert's
conjecture asserts that the maximum of φ on Kn occurs uniquely
at Jn=[1/n]n×n.
In this paper, we prove:
if A in Kn is partly decomposable then φ(A)<φ(Jn);
if the zeroes in A in Kn form a block then A is not
a φ-maximising matrix;
φ(A)<φ(Jn) unless
δ:=per(Jn)-per(A)≤ O(n4e-2n) and
|k-Σi inα ri| < √2δk,
|k-Σi inβ ci| < √2δk
and
Σi in α,j in β aij < k+√δk
for all sets α,β of k integers chosen from
{1,2,...,n}.