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.
@article{10_37236_12365,
author = {Yetong Sha and Huan Xiong},
title = {Proof of a conjecture of {Nath} and {Sellers} on simultaneous core partitions},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/12365},
zbl = {1536.05022},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12365/}
}
TY - JOUR
AU - Yetong Sha
AU - Huan Xiong
TI - Proof of a conjecture of Nath and Sellers on simultaneous core partitions
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/12365/
DO - 10.37236/12365
ID - 10_37236_12365
ER -
%0 Journal Article
%A Yetong Sha
%A Huan Xiong
%T Proof of a conjecture of Nath and Sellers on simultaneous core partitions
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/12365/
%R 10.37236/12365
%F 10_37236_12365
Yetong Sha; Huan Xiong. Proof of a conjecture of Nath and Sellers on simultaneous core partitions. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/12365
