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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