For a positive integer tβ₯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 sΓr rectangle.
Our main results are as follows. We first give a simple formula for the number
of partitions in the rectangle which 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 Ο 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,sββ maintaining a fixed aspect ratio, we again obtain a Gamma distribution
with the same shape parameter Ξ±=(tβ1)/2 and scale parameter Ξ²
that depends on the aspect ratio.Comment: 16 pages, 1 figure, improved exposition, references adde