  We illuminate the relation between the Bruhat order on the symmetric group
and structure constants (Littlewood-Richardson coefficients) for the cohomology
of the flag manifold in terms of its basis of Schubert classes. Equivalently,
the structure constants for the ring of polynomials in variables $x_1,x_2,...$
in terms of its basis of Schubert polynomials. We use combinatorial, algebraic,
and geometric methods, notably a study of intersections of Schubert varieties
and maps between flag manifolds. We establish a number of new identities among
these structure constants. This leads to formulas for some of these constants
and new results on the enumeration of chains in the Bruhat order. A new graded
partial order on the symmetric group which contains Young's lattice arises from
these investigations. We also derive formulas for certain specializations of
Schubert polynomials.
