Abaci structures of $(s,ms\pm 1)$-core partitions

Abstract

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 (originally proposed by T. Amdeberhan) on the enumeration of $(s,s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates the $(s,ms+1)$-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 $(s,ms\pm 1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.