The crank of partitions mod 8,9 and 10

Garvan, Frank G.

doi:10.1090/S0002-9947-1990-1012520-8

The crank of partitions mod $8,\;9$ and $10$
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by Frank G. Garvan

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

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

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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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