From partition identities to a combinatorial approach to explicit Satake inversion

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.