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 $\mathcal {A}(R,S)$ 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 $\mathcal {A}(R,S)$ are both extensions of the classical Bruhat order on Sn, the symmetric group of degree n. These two partial orders on $\mathcal {A}(R,S)$ are, in general, different. 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 on $\mathcal {A}(R,S)$ coincide.