Constructing integral matrices with given line sums

Criteria of valid line sum arrays for multidimensional matrices

“…Dias da Silva et al proved Theorem 3.1 by constructing a q-ary 2-matrix with a given line sum array [2]. Since we consider a generalization of this theorem to multidimensional matrices, we provide here a proof of Theorem 3.1 that employs the concept of multidimensional matrix.…”

A canonical construction for nonnegative integral matrices with given line sums

“…Using the definition of A S , we know that r ≥ s. We have to consider several cases: Case 1: a r,j = c r,j and a s,l = c s,l . Using the construction of C and the definition of So, we get (2). Suppose that r = s < m. Since a r,j + a r−1,j ≥ a r,l + a r−1,l > p and the fact that if c r−1,j = a r−1,j = p then c r−1,l = a r−1,l = p, we conclude that c r,j + c r−1,j ≥ c r,l + c r−1,l and c i,j = c i,l = p, for i = 1, .…”

“…More recently, using the domination relation between two partitions, Dias da Silva and Fonseca extended the Gale-Ryser theorem proving a necessary and sufficient condition for [2].) Let R and S be partitions of the same weight, and let p be a positive integer.…”

On the Largest Size of an Antichain in the Bruhat Order for

“…e.g. [2,3,4,5,7,8,9,16] and references therein). The Gale-Ryser Theorem, originally proved independently in [10] and [17], describing when (0, 1)matrices with given row and column sum vectors exist, lies at the heart of the classical combinatorial mathematics.…”