A formula for K-theory truncation Schubert calculus

Define a truncation rt(p of a polynomial p in $\{x_1,x_2,x_3,\dots\}$ as the polynomial with all but the first t variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulas for certain Schubert structure constants in K(Flags(ℂⁿ), in particular generalizing those studied by Kogan (2001) in which only cohomology was treated, and those studied by Buch (2002) on the Grassmannian case. The terms in the answer are computed using “marching” operations on permutation diagrams.