Overpartitions and Bressoud's conjecture. I (Q2671904)

Bressoud introduced partition functions \(A_j(\alpha_1,\ldots,\alpha_{\lambda};\eta,k,r;n)\) and \(B_j(\alpha_1,\ldots,\alpha_{\lambda};\eta,k,r;n)\) (\(j \in \{0,1\}\)) that each count integer partitions of \(n\) satisfying a list of congruence conditions/inequalities. He conjectured that they are always equal: \[ A_j(\alpha_1,\ldots,\alpha_{\lambda};\eta,k,r;n) = B_j(\alpha_1,\ldots,\alpha_{\lambda};\eta,k,r;n) \] for \(j \in \{0,1\}\), \((2k + j)/2 > r \geq \lambda \geq 0\) and \(n \geq 0\). This was recently resolved by \textit{S. Kim} [ibid. 325, 770--813 (2018; Zbl 1377.05014)] for \(j=1\), and the authors prove it in a subsequent paper for \(j=0\) based on the results of this paper. The main contribution of this paper are overpartition analogues \(\overline{A}_j\) and \(\overline{B}_j\) of \(A_j\) and \(B_j\). The authors prove a relationship between \(\overline{B}_0\) and \(B_1\) as well as a relationship between \(B_0\) and \(\overline{B}_1\). They also obtain an overpartition analogue of Bressoud's conjecture as well as new overpartition analogues of many classical partition theorems, such as Euler's partition theorem, the Rogers-Ramanujan-Gordon identities, and more.