Overpartitions and Bressoud's Conjecture, I

Abstract

In 1980, Bressoud conjectured a combinatorial identity Aj=Bj for j=0 or 1, where the function Aj counts the number of partitions with certain congruence conditions and the function Bj 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 B‾j which can be viewed as an overpartition analogue of the partition function Bj introduced by Bressoud. By means of Gordon markings, we build bijections to obtain a relationship between B‾1 and B0 and a relationship between B‾0 and B1. 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-Göllnitz-Gordon identities and the Bressoud-Göllnitz-Gordon identities.