New combinatorial interpretations of Ramanujan's partition congruences mod and

Author:
F. G. Garvan

Journal:
Trans. Amer. Math. Soc. **305** (1988), 47-77

MSC:
Primary 11P76; Secondary 05A17, 05A19

DOI:
https://doi.org/10.1090/S0002-9947-1988-0920146-8

MathSciNet review:
920146

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the number of unrestricted partitions of . The congruences referred to in the title are , and (, and , respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences and . These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence . In this paper we give a crank which not only gives a combinatorial interpretation of the congruence but also gives new combinatorial interpretations of the congruences and . 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 are encapsulated in this second identity. We give a number of relations for the crank of vector partitions and , as well as some new inequalities for the rank of ordinary partitions and . Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0920146-8

Keywords:
Partition congruences,
Dyson's rank,
crank,
Ramanujan's "lost" notebook

Article copyright:
© Copyright 1988
American Mathematical Society