Combinatorial expansions in K-theoretic bases

We study the class $\mathcal{C}$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-atoms, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $G{L}_n$, Grothendieck functions $\{G_{\lambda }\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal{C}$ is expanded in the $G$-basis or the basis dual to $\{G_{\lambda }\}$.