Geodesic
A weighted enumeration of maximal chains in the Bruhat order
Journal of Algebraic Combinatorics, Tome 15 (2002) no. 3, pp. 291-301.

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Summary: Given a finite Weyl group $W$ with root system $PHgr$, assign the weight $agrisinPHgr$ to each covering pair in the Bruhat order related by the reflection corresponding to $agr$. Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of $W$. Several variations and open problems are discussed.
Keywords: Bruhat order, Weyl group, root system 
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     author = {Stembridge, John R.},
     title = {A weighted enumeration of maximal chains in the {Bruhat} order},
     journal = {Journal of Algebraic Combinatorics},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2002__15_3_a0/}
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Stembridge, John R. A weighted enumeration of maximal chains in the Bruhat order. Journal of Algebraic Combinatorics, Tome 15 (2002) no. 3, pp. 291-301. http://geodesic.mathdoc.fr/item/JAC_2002__15_3_a0/