Given a curve $C$ the Brill-Noether variety $W^r_d(C)$ parameterizes line bundles on $C$ of degree $d$ and rank at least $r$. When $C$ is general in the moduli space $\mathcal{M}_g$ of smooth genus $g$ curves these varieties exhibit a number of ``desirable'' geometric properties and their dimension can be computed explicitly in terms of $g,r,$ and $d$. However, these varieties exhibit bizarre behaviour when one considers curves that are not general in $\mathcal{M}_g$. Our goal will be to understand how one can still study line bundles on these non-generic curves, called $k$-gonal curves. We begin with a study of the Brill-Noether varieties $W^r_d(C)$ and then consider a new variety $W^{\mu}(C)$ that parameterizes line bundles governed by the discrete invariant $\mu$.

Using machinery from tropical geometry and Berkovich spaces we may encode families of line-bundles as a special family of tableaux known as $k$-uniform displacement tableaux. We will discuss how $k$-uniform displacement tableaux on rectangular partitions parameterize $W^r_d(C)$. Furthermore, we will push this combinatorial analysis to a family of partitions known as $k$-cores to parameterize the varieties $W^{\mu}(C)$ explicitly in terms of $k$-uniform displacement tableaux.