Cores of partitions in rectangles

TL;DR

This paper studies the properties and distributions of $t$-cores of partitions within rectangular bounds, providing formulas, asymptotics, and limiting distributions relevant to combinatorics and representation theory.

Contribution

It introduces formulas and asymptotic results for $t$-cores of partitions in rectangles, extending previous work on partition distributions.

Findings

Number of $t$-core partitions in rectangles computed explicitly

Distribution of $t$-cores converges to a Gamma distribution as rectangle size grows

Distribution depends on aspect ratio in the large size limit

Abstract

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 $r \times s$ 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 $\rho$ 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 rate parameter $\beta$ that depends on the aspect ratio.