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 $P_i$