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Partition congruences and the Andrews-Garvan-Dyson crank
Author(s) -
Karl Mahlburg
Publication year - 2005
Publication title -
proceedings of the national academy of sciences
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.0506702102
Subject(s) - congruence relation , crank , partition (number theory) , mathematics , combinatorics , philosophy , geometry , cylinder
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.

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