Going beyond the partition theorem of Göllnitz - an exciting voyage.
by
Krishnaswami Alladi
University of Florida
Coauthors: George Andrews (Penn. State University), Alexander Berkovich (University of Florida)

One of the deepest results in the theory of partitions is the following theorem of Göllnitz (1967): The number of partitions of an integer n into distinct parts \equiv 2, 4, 5 (mod 6) equals the number of partitions of n in the form m₁+m₂+...+m_\nu such that m_\nu =/= 1 or 3 and m_i-m_i+1 >= 6 with strict inequality if m_i \equiv 0, 1, 3 (mod 6). In 1995, using the method of weighted words, Alladi-Andrews-Gordon obtained a three parameter refinement and generalization of Göllnitz's theorem. Subsequently, I obtained several reformulations of Göllnitz's theorem which explained its relationship with other fundamental results, yielded new weighted partition identities, and new combinatorial proofs. An intruiging question that Andrews raised in 1971 was whether there exist results that lie beyond Göllnitz's theorem. In November 1999, Alexander Berkovich and I discovered a remarkable q-series identity in four free parameters which reduces to the Alladi-Andrews-Gordon key identity for Göllnitz's theorem when one of the parameters is set equal to zero. The proof of this four parameter identity has just recently been completed in collaboration with Andrews and Berkovich, thereby settling this problem raised by Andrews about 30 years ago. This opens us several exciting avenues of exploration including the possibility of a new and different four parameter version of the Capparelli theorems that arose in a study of Lie algebras. In this talk the important ideas leading to the new four parameter identity will be explained, and some partition theorems that lie beyond Göllnitz's theorem will be stated. In a subsequent talk, Alexander Berkovich will outline the proof of the new four parameter q-series identity.

Date received: March 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-39.