  For a positive integer $t \geq 2$, 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 $s \times r$ 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 $\rho$ 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 $r,s \to
\infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution
with the same shape parameter $\alpha = (t-1)/2$ and scale parameter $\beta$
that depends on the aspect ratio.

Combinatorics;Probability