Beyond Göllnitz' theorem II: Arbitrarily many primary colors

Abstract

In 2003, Alladi, Andrews and Berkovich proved a four parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity.In this second paper, we state and bijectively prove an n(n+1)2-colored partition identity beyond Göllnitz' theorem for any number n of primary colors, along with the full set of the n(n−1)2 secondary colors as the product of two distinct primary colors, generalizing the identity proved in the first paper. Like the ten-colored partitions, our family of n(n+1)2-colored partitions satisfy some simple minimal difference conditions while avoiding forbidden patterns. Furthermore, the n(n+1)2-colored partitions have some remarkable properties, as they can be uniquely represented by oriented rooted forests which record the steps of the bijection.