Geodesic
Chains in the Bruhat order
Journal of Algebraic Combinatorics, Tome 29 (2009) no. 2, pp. 133-174.

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Summary: We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold $G/ B$. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees.
Keywords: keywords flag manifold, Schubert varieties, Bruhat order, saturated chains, harmonic polynomials, Grothendieck ring, Demazure modules, Schubert polynomials, flagged Schur polynomials, 312-avoiding permutations, kempf elements, vexillary permutations, Gelfand-tsetlin polytope, toric degeneration, parking functions, binary trees 
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     author = {Postnikov, Alexander and Stanley, Richard P.},
     title = {Chains in the {Bruhat} order},
     journal = {Journal of Algebraic Combinatorics},
     pages = {133--174},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2009__29_2_a5/}
}
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Postnikov, Alexander; Stanley, Richard P. Chains in the Bruhat order. Journal of Algebraic Combinatorics, Tome 29 (2009) no. 2, pp. 133-174. http://geodesic.mathdoc.fr/item/JAC_2009__29_2_a5/