We develop a geometric approach to the study of $(s,ms-1)$-core and $(s,ms+1)$-core partitions through the associated $ms$-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates $(s, ms+1)$-cores with distinct parts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.
@article{10_37236_6476,
author = {Rishi Nath and James A. Sellers},
title = {Abaci structures of \((s, ms\pm1)\)-core partitions},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/6476},
zbl = {1355.05039},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6476/}
}
TY - JOUR
AU - Rishi Nath
AU - James A. Sellers
TI - Abaci structures of \((s, ms\pm1)\)-core partitions
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6476/
DO - 10.37236/6476
ID - 10_37236_6476
ER -
%0 Journal Article
%A Rishi Nath
%A James A. Sellers
%T Abaci structures of \((s, ms\pm1)\)-core partitions
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6476/
%R 10.37236/6476
%F 10_37236_6476
Rishi Nath; James A. Sellers. Abaci structures of \((s, ms\pm1)\)-core partitions. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6476
