Flip posets of Bruhat intervals (Q1991425)

Summary: In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph \(B(u,v)\) of a Bruhat interval \([u,v]\) into rank posets \(P_{i}\) in a way that each \(P_{i}\) has a unique maximal chain that is rising under a reflection order. In the case where each \(P_{i}\) has rank three, the construction yields a combinatorial description of some terms of the complete \textbf{cd}-index as a sum of ordinary \textbf{cd}-indices of Eulerian posets obtained from each of the \(P_{i}\).