From The Cover: Partition congruences and the Andrews-Garvan-Dyson crank
Abstract
In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.
- Publication:
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Proceedings of the National Academy of Science
- Pub Date:
- October 2005
- DOI:
- 10.1073/pnas.0506702102
- Bibcode:
- 2005PNAS..10215373M
- Keywords:
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- MATHEMATICS