Beyond G\"ollnitz' Theorem II: arbitrarily many primary colors

TL;DR

This paper extends a partition identity beyond G"ollnitz's theorem to arbitrary many primary colors, providing bijective proofs and a novel forest representation, advancing the combinatorial understanding of multi-colored partitions.

Contribution

It introduces a generalized $rac{n(n+1)}{2}$-colored partition identity for any number of primary colors, with bijective proofs and a forest-based representation, extending prior work to more parameters.

Findings

Generalized partition identity for any number of primary colors

Bijective proof using forbidden patterns and minimal difference conditions

Unique forest representation of the colored partitions

Abstract

In $2003$, Alladi, Andrews and Berkovich proved a four-parameter partition identity lying beyond a celebrated identity of G\"ollnitz. 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 G\"ollnitz' 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.