New combinatorial interpretations of Ramanujan’s partition congruences mod $5,7$ and $11$

Abstract:

Let $p(n)$ denote the number of unrestricted partitions of $n$. The congruences referred to in the title are $p(5n + 4)$, $p(7n + 5)$ and $p(11n + 6) \equiv 0$ ($\bmod 5$, $7$ and $11$, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences $\bmod 5$ and $7$. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence $\bmod 11$. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence $\bmod 11$ but also gives new combinatorial interpretations of the congruences $\bmod 5$ and $7$. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone. Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan’s "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer’s paper. We note that all of Dyson’s conjectures $\bmod 5$ are encapsulated in this second identity. We give a number of relations for the crank of vector partitions $\bmod 5$ and $7$, as well as some new inequalities for the rank of ordinary partitions $\bmod 5$ and $7$. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.