Summary
New statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to thet-core of a partition. Usingq-series, some explicit formulas are given for the number of partitions which aret-cores. Some related questions for self-conjugate and distinct partitions are discussed.
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This work was partially supported by NSF grant DMS: 8700995
Oblatum 16-IX-1989
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Garvan, F., Kim, D. & Stanton, D. Cranks andt-cores. Invent Math 101, 1–17 (1990). https://doi.org/10.1007/BF01231493
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DOI: https://doi.org/10.1007/BF01231493