Beyond G\"ollnitz' Theorem I: A Bijective Approach

TL;DR

This paper introduces a bijective combinatorial approach to extend G"ollnitz's partition theorem beyond three primary and secondary colors, simplifying the proof and setting the stage for further generalizations.

Contribution

It provides a new bijective proof of a ten-colored partition identity that extends G"ollnitz's theorem without relying on complex $q$-series identities.

Findings

A bijective proof of a ten-colored partition identity

Equivalence established with previous $q$-series based identity

Simpler combinatorial formulation using forbidden patterns

Abstract

In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of G\"ollnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter $q$-series identity. In this paper we take a different approach. Instead of adding an eleventh quaternary color, we introduce forbidden patterns and give a bijective proof of a ten-colored partition identity lying beyond G\"ollnitz' theorem. Using a second bijection, we show that our identity is equivalent to the identity of Alladi, Andrews, and Berkovich. From a combinatorial viewpoint, the use of forbidden patterns is more natural and leads to a simpler formulation. In fact, in Part II of this series we will show how our method can be used to go beyond G\"ollnitz' theorem to any number of primary colors.