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

Richard A. Brualdi, Suk-Geun Hwang
2005 The Electronic Journal of Linear Algebra
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 presented that suggest such a
more » ... est such a determination might be very difficult for k ≥ 4. http://math.technion.ac.il/iic/ela II. Algorithm to Construct a Minimal Matrix in the Bruhat order for and let n ← k + r. (b) Else, q = 1, and let and let n ← k and k ← k − r. (c) Proceed recursively with the current values of n and k to determine X. Example. Let n = 18 and k = 11. The algorithm constructs the following minimal matrix in A(K, K).     Here we first construct (with 18 = 1 · 11 + 7), Then to construct the matrix X of order 11 with k = 11 − 7 = 4 (and 11 = 2 · 4 + 3), we construct Then to construct the matrix Y of order 4 + 3 = 7 with k = 4 (and 7 = 1 · 4 + 3), we construct J 3,4 O 3 Z J 4,3 .