The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. We reformulate the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions for skew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch's $K$-theoretic Littlewood-Richardson rule.