Identities of Littlewood-Richardson coefficients for Schubert polynomials and orders on S_n

For Schubert polynomials, the analogues of Littlewood-Richard\-son coefficients are expected to be related to the enumeration of chains in the Bruhat order on S_n. We refine this expectation in terms of certain suborders on the symmetric group associated to parabolic subgroups. Our main results are a number of new identities among these coefficients. For many of these identities, there is a companion result about the Bruhat order which we expect would imply the identity, were it known how to express these coefficients in terms of the Bruhat order. Our analysis leads to a new graded partial order on the symmetric group, results on the enumeration of chains in the Bruhat order, the determination of many of these constants, and formulas for a large class of specializations of the variables in a Schubert polynomial.