Overpartitions and Bressoud's conjecture, I

TL;DR

This paper introduces overpartition analogues of Bressoud's conjecture and related classical partition theorems, establishing bijections and relationships that extend known identities in partition theory.

Contribution

It develops overpartition analogues of Bressoud's conjecture and classical theorems, providing new bijections and relationships between overpartition functions and existing partition functions.

Findings

Established relationships between overpartition functions and classical partition functions.

Derived overpartition analogues of key partition theorems including Rogers-Ramanujan identities.

Extended classical identities to the overpartition setting using bijections.

Abstract

In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or $1$, where the function $A_j$ counts the number of partitions with certain congruence conditions and the function $B_j$ counts the number of partitions with certain difference conditions. Bressoud's conjecture specializes to a wide variety of well-known theorems in the theory of partitions. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim resolved Bressoud's conjecture for the case $j=1$. In this paper, we introduce a new partition function $\bar{B}_j$ which can be viewed as an overpartition analogue of the partition function $B_j$ introduced by Bressoud. By means of Gordon markings, we build bijections to obtain a relationship between $\bar{B}_1$ and $B_0$ and a relationship between $\bar{B}_0$ and $B_1$. Based on these former relationships, we further give overpartition analogues of many classical partition theorems including Euler's partition theorem, the Rogers-Ramanujan-Gordon identities, the Bressoud-Rogers-Ramanujan identities, the Andrews-G\"ollnitz-Gordon identities and the Bressoud-G\"ollnitz-Gordon identities.