Eliminating Human Insight: An Algorithmic Proof of Stembridge's TSPP Theorem

TL;DR

This paper introduces an automated, computer algebra-based proof of Stembridge's TSPP theorem, demonstrating potential for fully algorithmic proofs of complex combinatorial conjectures.

Contribution

It provides a fully automatic proof method for Stembridge's theorem using computer algebra, paving the way for proving the longstanding q-TSPP conjecture.

Findings

Automated proof of Stembridge's theorem achieved.

New computational methods developed for feasibility.

Potential to prove the q-TSPP conjecture with improved algorithms.

Abstract

We present a new proof of Stembridge's theorem about the enumeration of totally symmetric plane partitions using the methodology suggested in the recent Koutschan-Kauers-Zeilberger semi-rigorous proof of the Andrews-Robbins q-TSPP conjecture. Our proof makes heavy use of computer algebra and is completely automatic. We describe new methods that make the computations feasible in the first place. The tantalizing aspect of this work is that the same methods can be applied to prove the q-TSPP conjecture (that is a q-analogue of Stembridge's theorem and open for more than 25 years); the only hurdle here is still the computational complexity.