On the complete cd-index of a Bruhat interval
We study the non-negativity conjecture of the complete cd-index of a Bruhat interval as defined by Billera and Brenti. For each cd-monomial M we construct a set of paths, such that if a flip condition is satisfied, then the number of these paths is the coefficient of the monomial M in the complete cd-index. When the monomial contains at most one d, then the condition follows from Dyers proof of Cellinis conjecture. Hence the coefficients of these monomials are non-negative. We also relate the flip condition to shelling of Bruhat intervals.
complete cd-index shelling coxeter groups bruhat order