MIT Combinatorics Seminar

The Andrews-Garvan-Dyson Crank and Partition Congruences

Karl Mahlburg (MIT)

Wednesday, October 11, 2006   4:15 pm    Room 2-136


In 1944 Freeman Dyson proposed the existence of a crank function that would combinatorially explain the Ramanujan congruences for the partition function. Dyson's call wasn't answered until 1987, when Garvan and Andrews devised a combinatorial interpretation of some interesting $q$-series in Ramanujan's "Lost Notebook", and showed that this statistic decomposed the three congruences in a natural way. However, in 2000 Ono revolutionized the subject by proving the existence of infinite families of congruences, again raising the question of finding combinatorial explanations for the new congruences. The present work shows that the crank continues to act as a sort of "universal" statistic for partition congruences, and satisfies exactly the same general congruence properties as the partition function.