Cores of partitions in rectangles

Arvind Ayyer¹ ; Shubham Sinha²

¹Indian Institute of Science
²University of California San Diego

The electronic journal of combinatorics, Tome 31 (2024) no. 1

Résumé

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.

Détail

DOI : 10.37236/12022

Classification : 05A15, 05A16, 05A17, 60C05
Mots-clés : \(t\)-cores of partitions, uniformly random partition