A Bruhat order for the class of (0,1)-matrices with row sum vector R and column sum vector S

Abstract

Generalizing the Bruhat order for permutations (so for permutation matrices), a Bruhat order is defined for the class of m by n (0, 1)-matrices with a given row and column sum vector. An algorithm is given for constructing a minimal matrix (with respect to the Bruhat order)in such a class. This algorithm simplifies in the case that the row and column sums are all equal to a constant k. When k = 2 or k = 3, all minimal matrices are determined. Examples are presentedthat suggest such a determination might be very difficult for k ≥ 4.