ELA, Volume 10, pp. 280-290, December 2003, abstract.

On Nonnegative Matrices with Given Row and Column Sums

by

S.W. Drury, J.K. Merikoski, V. Laakso and T. Tossavainen

Let A be a nonnegative n-by-n matrix with row sums r_1,...,r_n 
and column sums c_1,...,c_n. Order them decreasingly:
r'_1 >= ...>= r'_n and c'_1 >= ... >= c'_n. The conjectures
su A^m =< (r_1c_1)^m/2 +...+ (r_nc_n)^m/2 and
su A^m =< (r'_1c'_1)^m/2 +...+ (r'_nc'_n)^m/2 are considered, 
where su B denotes the sum of the entries of a matrix B and 
m is a nonnegative integer.