One can employ elementary methods to demonstrate that $p(n) \leq p(n-1) + p(n-2)$ for $n \geq 2$. Recently, I showed that if certain restrictions are imposed on the partitions, the inequality becomes an equality.
\begin{equation} \notag \begin{aligned} &\ \ \ \ p(n\mid{\mathrm{parts}}\not\equiv{12,15,27}\ (\mathrm{mod}\ {27})) \\ &=p(n-1\mid{\mathrm{parts}}\not\equiv{6,21,27}\ (\mathrm{mod}\ {27}))) \\ &+p(n-2\mid{\mathrm{parts}}\not\equiv{3,24,27}\ (\mathrm{mod}\ {27}))). \\ \end{aligned} \end{equation}
I proved this identity using some $q$-series theory (https://arxiv.org/abs/2308.06289). Is it possible to find a combinatorial explanation for it?