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 Sym j pSym k V q for 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 GL 2 for these two cases.