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Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

Yetong Sha, Huan Xiong

Abstract

In 2016, Nath and Sellers proposed a conjecture regarding the precise largest size of ${(s,ms-1,ms+1)}$-core partitions. In this paper, we prove their conjecture. One of the key techniques in our proof is to introduce and study the properties of generalized-$β$-sets, which extend the concept of $β$-sets for core partitions. Our results can be interpreted as a generalization of the well-known result of Yang, Zhong, and Zhou concerning the largest size of $(s,s+1,s+2)$-core partitions.

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

Abstract

In 2016, Nath and Sellers proposed a conjecture regarding the precise largest size of -core partitions. In this paper, we prove their conjecture. One of the key techniques in our proof is to introduce and study the properties of generalized--sets, which extend the concept of -sets for core partitions. Our results can be interpreted as a generalization of the well-known result of Yang, Zhong, and Zhou concerning the largest size of -core partitions.
Paper Structure (9 sections, 21 theorems, 93 equations, 24 figures)

This paper contains 9 sections, 21 theorems, 93 equations, 24 figures.

Table of Contents

  1. Introduction
  2. $\beta$-sets and $s$-abacus diagrams
  3. The $\boldsymbol{\beta}$-sets of $\boldsymbol{(s,ms-1,ms+1)}$-core partitions
  4. Generalized-$\beta$-sets
  5. Preliminaries for the proofs
  6. Adjustments in one row
  7. Adjustments in different rows
  8. Adjustments in columns
  9. Discussions

Key Result

Theorem 1

The largest size of an $(s,ms-1,ms+1)$-core partition is There are two such maximal partitions; one's $\beta$-set is ${\mathcal{L}}_m(s)$ (please see Section 2 for the definition of $\beta$-sets and Section 3 for the definition of ${\mathcal{L}}_m(s)$), and the other one is the conjugate of the first one.

Figures (24)

  • Figure 1: The Young diagram and hook lengths of the partition $(6,3,2,1)$.
  • Figure 2: The Young diagram and hook lengths of the conjugation $(4,3,2,1,1,1)$ of the partition $(6,3,2,1)$.
  • Figure 3: $\mathcal{L}_3(5)$: The $\beta$-set of a maximal $(5,14,16)$-core partition.
  • Figure 4: The $\beta$-set of the other maximal $(5,14,16)$-core partition.
  • Figure 5: $\mathcal{L}_3(6)$: The $\beta$-set of a maximal $(6,17,19)$-core partition.
  • ...and 19 more figures

Theorems & Definitions (64)

  • Theorem 1: see Conjecture $57$ of NS2
  • Remark 2
  • Remark 3
  • Example 4
  • Definition 5: bergeolsOlsson
  • Remark 6
  • Example 7
  • Definition 8: size-counting function, see James
  • Lemma 9: see OlssonXiong1
  • Lemma 10: see OlssonXiong1
  • ...and 54 more