  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 concern with the case of to the classical flag manifold of Lie type A
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.
