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

2023/05/25 by Sha, Yetong, Xiong, Huan
#05A15 #05A17 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.15899

Abstract

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-β-sets, which extend the concept of β-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.

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