Balanced integer arrays: A matrix packing theorem

Abstract

Let t n be a vector of n positive integers that sum to 2 n − 1. Let u denote a vector of n or more positive integers that sum to n 2 , and call u , n-universal if for every possible choice of t 1 , t 2 ,…, t n , the components of the t i can be arranged in the successive rows of an n -row matrix (with 0 in each unused cell) so that u is the vector of column sums. It is shown that ( n ,…, n ) ( n times) is n -universal for every n . More generally, for odd n , any choice of t 1 , t 3 ,…, t n can be placed in rows so that the column sums are ( n , n −1,…, 2, 1); for even n , any choice of t 2 , t 4 ,…, t n can be placed in rows so that the column sums are ( n , n −1,…, 2, 1). Hence, any u that can be obtained from the sum of two rows whose nonzero components are, respectively, n , n −1,…, 2, 1 and n −1, n −2,…, 2, 1 (in any order, with 0's elsewhere) is n -universal. The problem examined is closely related to the graph conjecture that trees on 2, 3,…, n + 1 vertices can be superposed to yield the complete graph on n + 1 vertices.