oswald kluge

Given two positive integers a and b, find an expression for all m times
n matrices N with non-negative integer coefficients such that

N*D=A and E*N=B,

whereas D=(1,2,3,..,n)^T, E=(1,2,3,..,m), A=(a,..,a)^T and
B=(b,..,b).

Have fun!

Oswald

israel

Of course we must have E A = E N D = B D, i.e. a m (m+1) = b n (n+1).
Then the entries N_{1,j} and N_{i,1} can be expressed in terms of the others:
N_{1,j} = b - sum_{i=2}^m i N_{i,j} for j > 1
N_{i,1} = a - sum_{j=2}^n j N_{i,j} for i > 1
N_{1,1} = b - sum_{i=2}^m i N_{i,1}
= b - a (m+2)(m-1)/2 + sum_{i=2}^m sum_{j=2}^n i j N_{i,j}

If the N_{i,j} for i=2..m, j=2..n are integers, then so are the N_(1,j)
and N_(i,1). All that remains is to ensure that these are nonnegative.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Thanks for your reply! Is there more information about the N_{i,j} if
the N's also have to hold

(1,...,1)*N*(1,..,1)=k

for a fixed positive integer k?

Best,
O. K.