The crank of partitions mod and

Author:
Frank G. Garvan

Journal:
Trans. Amer. Math. Soc. **322** (1990), 79-94

MSC:
Primary 11P83; Secondary 05A17, 11P81

DOI:
https://doi.org/10.1090/S0002-9947-1990-1012520-8

MathSciNet review:
1012520

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently new combinatorial interpretations of Ramanujan's partition congruences modulo , and were found. These were in terms of the crank. A refinement of the congruence modulo is proved. The number of partitions of with even crank is congruent to 0 modulo . The residue of the even crank modulo divides these partitions into five equal classes. Other relations for the crank modulo , and 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:
https://doi.org/10.1090/S0002-9947-1990-1012520-8

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