Rank, crank, and mex: New connections between partition statistics

About 100 years ago, Ramanujan proved certain patterns in the counts of integer partitions, but not in a way that fully "explained" them. A young Freeman Dyson wrote in a somewhat cheeky 1944 article that a new notion he called the rank of a partition explained some of the patterns of partition counts—without proving it—and that something called the crank should explain the rest—without defining crank! Everything he proposed was eventually proven by others to be correct. The new part of the story is recent work of the speaker and James Sellers that explains crank, whose definition is somewhat tricky, in terms of the minimal excluded part ("mex") of integer partitions. This allows us to improve and simplify a recent result in the Ramanujan Journal.

Brian Hopkins