Séminaire Lotharingien de Combinatoire, 84B.1 (2020), 12 pp.

Isaac Konan

Beyond Göllnitz' Theorem I: A Bijective Approach

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 Göllnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter q-series identity. In this extended abstract, 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 Göllnitz' 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 following the full paper, we show how our method can be used to go beyond Göllnitz' theorem to any number of primary colors.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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