Note on a partition limit theorem for rank and crank. (11th January 2013) - Diaconis, Persi; Janson, Svante; Rhoades, Robert C.
Abstract : If λ is a partition of n, then the rank rk(λ) is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, in K. Bringmann, K. Mahlburg, and R. C. Rhoades ( Bull. Lond. Math. Soc., 43 (2011) 661–672), it is shown thatrk ( λ ) / 6 n has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of β ( T ), where β ( t ) is standard Brownian motion and T is the first time that an independendent 3‐dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.