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

Rishi Nath, James A. Sellers

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.

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

Abstract

We develop a geometric approach to the study of -core and -core partitions through the associated -abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of and -core partitions with distinct parts. It also enumerates the -cores with distinct parts. Furthermore, we calculate the weight of the -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 -abaci of maximal -cores can be built up from -abaci of -cores in an elegant way.

Paper Structure

This paper contains 10 sections, 41 theorems, 20 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Partitions and abacus diagrams
  3. $s$-core and simultaneous $(s,t)$-core partitions
  4. Simultaneous $(s,t)$-cores with distinct parts
  5. Simultaneous $(s,ms-1,ms+1)$-core partitions
  6. $s$-abaci of $(s,s\pm1)$-cores
  7. $ms$-abaci of $(s,ms\pm1)$-cores
  8. $ms$-abaci of $(s,ms\pm1)$-cores with distinct parts
  9. the $ms$-abacus of the longest $(s,ms-1,ms+1)$-core
  10. $ms$-abaci of self-conjugate $(s,ms\pm1)$-core partitions with distinct parts

Key Result

Lemma \oldthetheorem

Let $X$ be a bead-set of a partition $\lambda.$ The size of the part $\lambda_{\alpha}$ of $\lambda$ corresponding to the bead $x'\in X$ is the number of spacers to the left of the bead, that is, $\lambda_{\alpha}=|y\not\in X:y<x'|.$

Figures (8)

  • Figure 1: ${\mathcal{A}(5)}$
  • Figure 2: ${\mathcal{B}_0(5)}$: $5$-abacus of a $(5,9)$-core
  • Figure 3: ${\mathcal{B}_1(5)}$: $5$-abacus of the maximal $(4,5)$-core
  • Figure 4: ${\mathcal{E}_3^-}(5)$: $15$-abacus of the maximal $(5,14)$-core
  • Figure 5: ${\mathcal{E}_3^+}(5)$: $15$-abacus of the maximal $(5, 16)$-core
  • ...and 3 more figures

Theorems & Definitions (89)

  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem: $s$-abacus
  • Definition \oldthetheorem: $s$-abacus position
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem: J. Olsson and D. Stanton, Theorem 4.1, O-S
  • ...and 79 more