(0, 1)-Matrices, Discrepancy and Preservers

Abstract

Let m and n be positive integers, and let R = (r₁, . . . , r_m) and S = (s₁, . . . , s_n) be nonnegative integral vectors. Let A(R,S) be the set of all m × 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 × n (0, 1)-matrix, where for each i, the ith row of F(R,S) consists of r_i 1’s followed by (n−r_i) 0’s. Let A ∈ 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 linear operators mapping m × n matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when m = n, the transpose mapping.