A monoid for the universal k-Bruhat order
Nantel Bergeron and Frank Sottile
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 Sym$_\infty$, the universal $k$-Bruhat order. Here we present a monoid M for this order. We show that M is analogous to the nil-Coxeter monoid for the weak order on Sym$_\infty$. For this, we develop the theory of reduced sequences for M. We use these sequences 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.