Geodesic
On the complete cd-index of a Bruhat interval
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 527-541.

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Summary: 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 Dyer's proof of Cellini's conjecture. Hence the coefficients of these monomials are non-negative. We also relate the flip condition to shelling of Bruhat intervals.
Keywords: complete cd-index, Coxeter groups, Bruhat order, shelling 
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     author = {Karu, Kalle},
     title = {On the complete cd-index of a {Bruhat} interval},
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Karu, Kalle. On the complete cd-index of a Bruhat interval. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 527-541. http://geodesic.mathdoc.fr/item/JAC_2013__38_3_a9/