Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

In 2016, Nath and Sellers proposed a conjecture regarding the precise largest size of $(s,ms-1,ms+1)$-core partitions. In this paper, we prove their conjecture. One of the key techniques in our proof is to introduce and study the properties of generalized-$\beta$-sets, which extend the concept of $\beta$-sets for core partitions. Our results can be interpreted as a generalization of the well-known result of Yang, Zhong, and Zhou concerning the largest size of $(s,s+1,s+2)$-core partitions.