A bijection related to Bressoud's conjecture

Bressoud introduced the partition function $B({\alpha }_1,\dots ,{\alpha }_{\lambda };\eta ,k,r;n)$, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition function $B({\alpha }_1,\dots ,{\alpha }_{\lambda };\eta ,k,r;n)$ in multi-summation form. In this article, we introduce a bijection related to Bressoud's conjecture. As an application, we give the proof of a companion to the Göllnitz-Gordon identities.