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 $\frac{n(n+1)}{2}$-colored partition identity beyond Göllnitz' theorem for any number n of primary colors, along with the full set of the $\frac{n(n-1)}{2}$ 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 $\frac{n(n+1)}{2}$-colored partitions satisfy some simple minimal difference conditions while avoiding forbidden patterns. Furthermore, the $\frac{n(n+1)}{2}$-colored partitions have some remarkable properties, as they can be uniquely represented by oriented rooted forests which record the steps of the bijection.

2003 年，Alladi、Andrews 和 Berkovich 证明了一个四参数分区恒等式，它超越了著名的 Göllnitz 恒等式。从那时起，将他们的工作扩展到五个或更多参数一直是一个悬而未决的问题。在这两篇论文的第一部分中，我们通过对他们的结果进行重新表述的双射证明，朝着这个方向迈出了第一步。我们引入了禁止模式，双射证明了十色分区恒等式，然后通过另一个双射将我们的恒等式与 Alladi-Andrews-Berkovich 恒等式相关联。

在第二篇论文中，我们陈述并双射地证明了一个$\frac{n(n+1)}{2}$- 超出 Göllnitz 定理的彩色分区恒等式，适用于任意数量n的原色，以及完整的$\frac{n(n-1)}{2}$二次色作为两种不同原色的产物，概括了第一篇论文中证明的同一性。就像十色隔断，我们家$\frac{n(n+1)}{2}$-colored 分区满足一些简单的最小差异条件，同时避免禁止模式。此外，该$\frac{n(n+1)}{2}$彩色分区具有一些显着的特性，因为它们可以由记录双射步骤的定向有根森林唯一地表示。