  We address a unification of the Schubert calculus problems solved by [A. Buch
'02] and [A. Knutson-T. Tao '03]. That is, we prove a combinatorial rule for
the structure coefficients in the torus-equivariant K-theory of Grassmannians
with respect to the basis of Schubert structure sheaves. We thereby deduce the
conjectural rule of [H. Thomas-A. Yong '13] for the same coefficients. Both
rules are positive in the sense of [D. Anderson-S. Griffeth-E. Miller '11] (and
moreover in a stronger form). Our work is based on the combinatorics of genomic
tableaux and a generalization of [M.-P. Schutzenberger '77]'s jeu de taquin.
