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Overpartitions and Bressoud's conjecture, II

Thomas Y. He, Kathy Q. Ji, Alice X. H. Zhao

Abstract

The main objective of this paper is to present an answer to Bressoud's conjecture for the case $j=0$, resulting in a complete solution to the conjecture. The case for $j=1$ has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function $B_0$ and the overpartition function $\overline{B}_1$, we found that the proof of Bressoud's conjecture for the case $j=0$ is equivalent to establishing an overpartition analogue of the conjecture for $j=1$. By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for $j=1$, which eventually enables us to confirm Bressoud's conjecture for the case $j=0$.

Overpartitions and Bressoud's conjecture, II

Abstract

The main objective of this paper is to present an answer to Bressoud's conjecture for the case , resulting in a complete solution to the conjecture. The case for has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function and the overpartition function , we found that the proof of Bressoud's conjecture for the case is equivalent to establishing an overpartition analogue of the conjecture for . By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for , which eventually enables us to confirm Bressoud's conjecture for the case .

Paper Structure

This paper contains 10 sections, 22 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Over-bre-1bbb']}
  3. Proof of Theorem \ref{['Over-bre-1a']}
  4. Proof of Theorem \ref{['Over-bre-1a']} with the aid of Theorem \ref{['lambda-r']}
  5. The outline of the proof of Theorem \ref{['lambda-r']}
  6. The (reverse) Gordon marking and the forward (backward) move
  7. The $(k-1)$-addition and the $(k-1)$-subtraction
  8. The $(k-1)$-insertion and the $(k-1)$-separation
  9. Proof of Theorem \ref{['lambdathm']}
  10. Example

Key Result

Theorem 1.6

he-ji-zhao For $k\geq r\geq \lambda\geq0$ and $k>\lambda$,

Theorems & Definitions (35)

  • Definition 1.1: Bressoud
  • Definition 1.2: Bressoud
  • Conjecture 1.3: Bressoud
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: Bressoud
  • Theorem 1.9
  • Theorem 1.10
  • ...and 25 more