Structure constants for the multiplication of Schubert polynomials by
Schur symmetric polynomials are known to be related to the enumeration of
chains in a new partial order on $S_\infty$, the Grassmann-Bruhat
order. Here we present a monoid M for this order analogous to the
nil-Coxeter monoid for the weak order on $S_\infty$. For this, we develop
the theory of reduced decompositions for M. We use this to give a
combinatorial description of the structure constants above. We also give a
combinatorial proof of some of the symmetry relations satisfied by these
constants.