Cores of Partitions in Rectangles

For a positive integer t ≥ 2, the t-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of t-cores of partitions contained in an {r× s} rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves t-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose t-cores are a fixed partition ρ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the t-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer n (Ann. Appl. Prob. 2023). In particular, we show that in the limit as r,s → ∞ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter α = (t-1)/2 and rate parameter β that depends on the aspect ratio.