A canonical construction for nonnegative integral matrices with given line sums

Let p   be a positive integer and let A(p)(R,S)A(p)(R,S) be the class of nonnegative integral matrices with entries less than or equal to p, with row–sum partition R, and column–sum partition S.In this paper we state a new necessary and sufficient condition for A(p)(R,S)≠∅A(p)(R,S)≠∅. This condition generalizes the well known Gale–Ryser theorem. We also present a canonical construction for matrices in A(p)(R,S)A(p)(R,S).