  In work with A. Yong, the author introduced genomic tableaux to prove the
first positive combinatorial rule for the Littlewood-Richardson coefficients in
torus-equivariant $K$-theory of Grassmannians. We then studied the genomic
Schur function $U_\lambda$, a generating function for such tableaux, showing
that it is non-trivially a symmetric function, although generally not
Schur-positive. Here we show that $U_\lambda$ is, however, positive in the
basis of fundamental quasisymmetric functions. We give a positive combinatorial
formula for this expansion in terms of gapless increasing tableaux; this is,
moreover, the first finite expression for $U_\lambda$. Combined with work of A.
Garsia and J. Remmel, this yields a compact combinatorial (but necessarily
non-positive) formula for the Schur expansion of $U_\lambda$.
