  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 $GL_n$,
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 $\{G_\lambda\}$.
