Overpartitions and Bressoud's conjecture, II

TL;DR

This paper proves Bressoud's conjecture for the case j=0 by establishing an overpartition analogue for j=1, completing the proof for both cases through generalized methods.

Contribution

It provides the first complete proof of Bressoud's conjecture for j=0 by linking it to overpartition analogues and extending Kim's method.

Findings

Confirmed Bressoud's conjecture for j=0.

Established an overpartition analogue for j=1.

Extended Kim's method to overpartition contexts.

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 $j=0$.