Abaci Structures of $(s, ms\pm1)$-Core Partitions

Rishi Nath,James A Sellers

doi:10.37236/6476

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 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.

Highlights

A partition has distinct parts if and only if its minimal bead-set X satisfies the following property: if x, y ∈ X and x > y, x − y = 1
The combination of Lemma 4.1 and Theorem 3.6 allow us to study the abaci of certain simultaneous core partitions with distinct parts
(2) The minimal ms-abacus M− of any (s, ms − 1)-core partition with distinct parts will be a sub-abacus of Em−(s) consisting of beads taken only from its first row

Summary

Introduction

A partition has distinct parts if and only if its minimal bead-set X satisfies the following property: if x, y ∈ X and x > y, x − y = 1. The combination of Lemma 4.1 and Theorem 3.6 allow us to study the abaci of certain simultaneous core partitions with distinct parts.

Results

Conclusion