Classes of (0,1)-matrices Where the Bruhat Order and the Secondary Bruhat Order Coincide

Abstract

Given two nonincreasing integral vectors R and S, with the same sum, we denote by A(R, S) the class of all (0, 1)-matrices with row sum vector R, and column sum vector S. The Bruhat order and the Secondary Bruhat order on A(R, S) are both extensions of the classical Bruhat order on S n , the symmetric group of degree n. These two partial orders are not, in general, the same. In this paper we prove that if R = (2, 2, . . . , 2) or R = (1, 1, . . . , 1), then the Bruhat order and the Secondary Bruhat order coincide on A(R, S).

Extremal matrices for the Bruhat-graph order

“…If A 1 = A 2 , we write A 1 ≺ B A 2 . The Bruhat order for (0, 1)-matrices in the classes A(R) is a pre-order that has been extensively investigated [1][2][3][4][5][6][7].…”

On a conjecture concerning the Bruhat order

“…Many researchers have tried to prove this conjecture and consequently, other notions arose linked to the Bruhat orders, [5,6,7,8,9,10]. One of them was the notion of an inversion in a (0, 1)-matrix.…”

The Bruhat order on classes of isotopic Latin squares

“…The Bruhat order on (0, 1)-matrices is receiving the attention of many researchers, [4,6,10,11,12,13,15,16,17,18,19]. In the recent years several authors have taken Brualdi and Hwang's ideas, and extended the Bruhat order to other classes of matrices than (0, 1)-matrices: the Bruhat order has been studied on the class of tournament matrices with a given score vector, [8], on the class of alternating sign matrices, [9], and on the class of doubly stochastic matrices, [7].…”