Partition congruences and the Andrews-Garvan-Dyson crank.
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.
Mahlburg
Karl
K
Department of Mathematics, University of Wisconsin, 418 Van Vleck Hall, E B, 480 Lincoln Drive, Madison, WI 53706, USA. mahlburg@math.wisc.edu