讲座内容：

Amdeberhan's conjectures concerning the enumeration, average size, and largest size of strict $(n,n+1)$-core partitions have stimulated extensive research in this area. Nevertheless, the investigation of random core partitions remains relatively underdeveloped. In this talk, we first present several polynomiality results and asymptotic formulas for the moments of sizes of random strict $(n, dn\pm 1)$-core partitions. We further establish the asymptotic normality of their sizes. Analogous results are also derived for random (strict) $n$-core partitions and random self-conjugate $n$-core partitions with bounded perimeters. These findings resolve several conjectures posed by Zaleski and stand in contrast to the asymptotic behavior derived by Even-Zohar in 2022 for the size of a random $(s,t)$-core partition with coprime $s$ and $t$, which converges in distribution to Watson’s $U^2$ distribution. These are joint works with Wenston J.T. Zang, Yetong Sha, and Jiange Li.