Abaci structures of $(s,ms\pm1)$-core partitions

TL;DR

This paper introduces a geometric approach using abaci to analyze and enumerate specific classes of core partitions, providing new proofs and extending enumeration results for partitions with distinct parts.

Contribution

It develops a novel geometric framework with abaci to study $(s,ms\pm1)$-core partitions, offering new proofs and extending enumeration results.

Findings

Enumerates $(s,ms\pm1)$-core partitions with distinct parts.

Calculates the weight of the largest $(s,ms-1,ms+1)$-core partition.

Provides a method to build $ms$-abaci from $s$-abaci for maximal cores.

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 $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.