  The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians induces
a K-theory pullback that defines the splitting coefficients. We geometrically
explain an identity from [Buch '02] between the splitting coefficients and the
Schubert structure constants for products of Schubert structure sheaves. This
is related to the topic of product and splitting coefficients for Schubert
boundary ideal sheaves. Our main results extend jeu de taquin for increasing
tableaux [Thomas-Yong '09] by proving transparent analogues of
[Sch\"{u}tzenberger '77]'s fundamental theorems on well-definedness of
rectification. We then establish that jeu de taquin gives rules for each of
these four kinds of coefficients.
