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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The crank of partitions mod $ 8,\;9$ and $ 10$


Author: Frank G. Garvan
Journal: Trans. Amer. Math. Soc. 322 (1990), 79-94
MSC: Primary 11P83; Secondary 05A17, 11P81
MathSciNet review: 1012520
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Abstract: Recently new combinatorial interpretations of Ramanujan's partition congruences modulo $ 5$, $ 7$ and $ 11$ were found. These were in terms of the crank. A refinement of the congruence modulo $ 5$ is proved. The number of partitions of $ 5n + 4$ with even crank is congruent to 0 modulo $ 5$. The residue of the even crank modulo $ 10$ divides these partitions into five equal classes. Other relations for the crank modulo $ 8$, $ 9$ and $ 10$ are also proved. The dissections of certain generating functions associated with these results are calculated. All of the results are proved by elementary methods.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1012520-8
PII: S 0002-9947(1990)1012520-8
Keywords: Congruences, crank, dissections, generating functions, Macdonald identities, modular functions, partitions, quadratic forms, symmetry group, theta functions, Ramanujan
Article copyright: © Copyright 1990 American Mathematical Society