Partition Congruences and the Andrews-Garvan-Dyson Crank

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Partition Congruences and the Andrews-Garvan-Dyson Crank is a paper by Karl Mahlburg that was published in Proceedings of the National Academy of Sciences.

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, George Andrews and Garvan successfully found such a function [3, 6], and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7 and 11. This note announces the proof [8] of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

The full paper can be found at the links below.

[edit] External links

• Abstract. Proceedings of the National Academy of Science. Retrieved on March 31, 2007.
• Karl Mahlburg (2005). "Partition Congruences and the Andrews-Garvan-Dyson Crank" (PDF). Proceedings of the National Academy of Sciences 102 (43): 15373-76.