On maximum chains in the Bruhat order of A(n,2)

Let A(R,S) denote the class of all matrices of zeros and ones with row sum vector R and column sum vector S. We introduce the notion of an inversion in a (0,1)-matrix. This definition extends the standard notion of an inversion of a permutation, in the sense that both notions agree on the class of permutation matrices. We prove that the number of inversions in a (0,1)-matrix is monotonic with respect to the secondary Bruhat order of the class A(R,S). We apply this result in establishing the maximum length of a chain in the Bruhat order of the class A(n,2) of (0,1)-matrices of order n in which every row and every column has a sum of 2. We give algorithmic constructions of chains of maximum length in the Bruhat order of A(n,2).