(0,1)-matrices and Discrepancy

Abstract

 Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $A\in A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $\ell$, then the discrepancy of the transpose of $A$ is at least $\frac{\ell}{2}$ and at most $2\ell$. These bounds are tight.