Flip posets of Bruhat intervals

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 cd-index as a sum of ordinary cd-indices of Eulerian posets obtained from each of the P-i.