# A partition congruence modulo 13

In the paper "Note on certain modular relations considered by Messrs Ramanujan, Darling and Rogers" (Proceedings of London Mathematical Society (1922) s2-20 (1): 408-416) Mordell gives proofs of the identities \begin{align} \sum_{n = 0}^{\infty}p(5n + 4)q^{n} &= 5\frac{\{(1 - q^{5})(1 - q^{10})(1 - q^{15})\cdots\}^{5}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{6}}\tag{1}\\ \sum_{n = 0}^{\infty}p(7n + 5)q^{n} &= 7\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{3}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{4}}\notag\\ &\,\,\,\,+ 49q\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{7}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{8}}\tag{2} \end{align} based on theory of elliptic modular functions. He uses a slightly old notation $$\Delta(\omega_{1}, \omega_{2}) = \left(\frac{2\pi}{\omega_{2}}\right)^{12}q\prod_{n = 1}^{\infty}(1 - q^{n})^{24}$$ where $q = e^{2\pi i\omega}, \omega = \omega_{1}/\omega_{2}$.

Clearly both the series given by Ramanujan lead to partition congruences modulo $5, 5^{2}, 7, 7^{2}$. Mordell shows that these results are equivalent to $$\sum_{j = 0}^{4}\left(\frac{\Delta(5\omega_{1}, \omega_{2})}{\Delta\{(\omega_{1} + j\omega_{2})/5, \omega_{2}\}}\right)^{1/24} = 5\left(\frac{\Delta(5\omega_{1}, \omega_{2})}{\Delta(\omega_{1}, \omega_{2})}\right)^{1/4}\tag{3}$$ and \begin{align} \sum_{j = 0}^{6}\left(\frac{\Delta(7\omega_{1}, \omega_{2})}{\Delta\{(\omega_{1} + j\omega_{2})/7, \omega_{2}\}}\right)^{1/24} &= 7\left(\frac{\Delta(7\omega_{1}, \omega_{2})}{\Delta(\omega_{1}, \omega_{2})}\right)^{1/6}\notag\\ &\,\,\,\,+ 49\left(\frac{\Delta(7\omega_{1}, \omega_{2})}{\Delta(\omega_{1}, \omega_{2})}\right)^{1/3}\tag{4} \end{align} Mordell goes on to mention that identities like $(3), (4)$ don't exist if we replace replace $5, 7$ by $11$. On the other hand he says that there is similar identity for modulo $13$.

This seems very weird because we do have a congruence identity $p(11n + 6) \equiv 0\pmod{11}$ and we don't have a similar partition congruence $\pmod{13}$. What Mordell probably means is that we do not have an expansion like $(1)$ or $(2)$ for $p(11n + 6)$ and at the same time we do have some formula like $(1), (2)$ for $p(13n + a)$ for some integer $a, 0 < a < 13$.

Is my understanding mentioned in last paragraph correct? If so, do we have an identity like $(1), (2)$ available in literature for modulo $13$? Any references regarding such an identity and its proof would be really helpful.