  In this paper, we provide combinatorial proofs for certain partition
identities which arise naturally in the context of Langlands' beyond endoscopy
proposal. These partition identities motivate an explicit plethysm expansion of
$\mathrm{Sym}^j(\mathrm{Sym}^kV)$ for $\mathrm{GL}_2$ in the case $k=3$. We
compute the plethysm explicitly for the cases $k=3, 4$. Moreover, we use these
expansions to explicitly compute the basic function attached to the symmetric
power $L$-function of $\mathrm{GL}_2$ for these two cases.
