Bruhat order of tournaments

Abstract

We extend the Bruhat order on the set Sn of permutations (permutation matrices) of {1,2,…,n} and its generalization to classes A(R,S) of (0,1)-matrices with row sum vector R and column sum vector S, to a Bruhat order on classes T(R) of tournaments with score vector R. As in the case of the Bruhat order on A(R,S), there are two possible Bruhat orders where one is a refinement of the other. We characterize the cover relation for one of these orders. For a special family of score vectors, we show these Bruhat orders are isomorphic to the partially ordered set of all subsets of a set.