Beyond Gollnitz' Theorem I: Abijective approach

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:180

Article

Konan, Isaac¹ 

[1] Univ Paris, IRIF, Batiment Sophie Germain,Case Courrie 7014, F-75205 Paris 13, France

关键词: Rogers-Ramanujan type identities;    Weighted words;    Gollnitz' identity;    Bijections;   

DOI  :  10.1016/j.jcta.2021.105426

来源: Elsevier

【 摘 要 】

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 Gollnitz, 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 Gollnitz' 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 Gollnitz' theorem to any number of primary colors. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free