Schubert Products for Permutations with Separated Descents.

We say that two permutations $\pi$ and $\rho$ have separated descents at position $k$ if $\pi$ has no descents before position $k$ and $\rho$ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents. Our approach uses generalizations of Sch\"utzenberger's jeu de taquin algorithm and the Edelman-Greene correspondence via bumpless pipe dreams.