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 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 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 of this series we will show how our method can be used to go beyond Göllnitz' theorem to any number of primary colors.

Beyond Göllnitz' Theorem II: arbitrarily many primary colors

“…Note that when d = 0, we recover Alladi-Andrews-Gordon's refinement of Göllnitz' identity (see [10] for more details). Their main tool was an intricate q-series identity.…”

“…Later, Alladi-Andrews-Gordon [2] found a refinement of Göllnitz' identity, with the use of weighted words with three primary colors a, b, c and three secondary colors ab, ac, bc, which indeed implies the refinement of Schur's identity. Further explanation of these two refinements is given in the first part of this series [10].…”

“…In part I of this series [10], we showed an equivalent version of Theorem 1.1. In fact, we supposed that the parts occur in only primary colors a, b, c, d and secondary colors ab, ac, ad, bc, bd, cd, and are ordered as in (1.2) by omitting quaternary parts:…”

“…Example 1.4. For example, with n = 49, the partitions of the first kind are (35,10,4), (34,11,4), (28,11,10), (23,22,4), (23,16,10), (22,16,11) and (16,11,10,8,4) and the partitions of the second kind are (35,14), (34,15), (33,16), (45,4), (39, 10), (38,11) and (27, 18, 4) •…”