Combinatorics Seminar: Partition bijections: beyong Gollnitz theorem

Seminar | December 2 | 12:10-1 p.m. | 939 Evans Hall | Canceled

 Isaac Konan, Universite de Paris

 Department of Mathematics

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 talk, we present 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, our method allows us to state a theorem beyond Göllnitz’ identiy, starting from a set of n primary colors and using the n(n-1)/2 commutative secondary colors different from the squares.