[Paper Review] A Proof of George Andrews' and Dave Robbins' q-TSPP Conjecture (modulo a finite amount of routine calculations)

Manuel Kauers, Christoph Koutschan+1 more|ArXiv.org|Aug 5, 2008

Advanced Combinatorial Mathematics12 references6 citations

TL;DR

This paper presents a proof of the q-TSPP conjecture—formulating a q-analog of the generating function for totally symmetric plane partitions—using a symbolic computation approach based on determinant evaluation and certificate-based verification. The key result is a rigorous proof modulo a finite number of routine calculations, leveraging holonomic functions and computer algebra systems to validate a determinant identity proposed by Soichi Okada.

ABSTRACT

In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre Leroux himself, who talked about his joint work with Xavier Viennot[LV], on solving differential equations combinatorially! During the problem session of that very same colloque, chaired by Pierre Leroux, Richard Stanley raised some intriguing problems about the enumeration of plane partitions, that he later expanded into a fascinating article[Sta1]. Most of these problems concerned the enumeration of symmetry classes of plane partitions, that were discussed in more detail in another article of Stanley[Sta2]. All of the conjectures in the latter article have since been proved (see Dave Bressoud's modern classic[B]), except one, that, so far, resisted the efforts of the greatest minds in enumerative combinatorics. It concerns the proof of an explicit formula for the q-enumeration of totally symmetric plane partitions, conjectured independently by George Andrews and Dave Robbins([Sta2],[Sta1](conj. 7), [B](conj. 13)). In this tribute to Pierre Leroux, we describe how to prove that last stronghold.

Motivation & Objective

• To resolve the longstanding q-TSPP conjecture, a q-analog of the generating function for totally symmetric plane partitions, which had resisted proof for decades.
• To demonstrate that the conjecture can be reduced to verifying a specific determinant identity using symbolic computation techniques.
• To provide a framework for transforming combinatorial determinant identities into formally certifiable problems via holonomic functions and recurrence relations.
• To establish a pathway toward a fully rigorous proof using modern computer algebra systems, even if currently reliant on computational feasibility.

Proposed method

• Reduces the q-TSPP conjecture to proving a determinant identity via Soichi Okada’s reduction, which connects the generating function to a matrix determinant.
• Employs a certificate-based method for determinant evaluation, where a function $ B(n,j) $ is proposed to satisfy three conditions: Soichi’s recurrence, normalization, and Okada’s evaluation identity.
• Uses the holonomic ansatz to describe $ B(n,j) $ as a holonomic sequence, enabling symbolic manipulation and recurrence-based verification.
• Translates the determinant verification into constant-term identities in a bivariate generating function $ f_n(x) = \sum_j C(n,j)x^j $, where $ C(n,j) = B(n,n-j) $.
• Reformulates the verification conditions into differential-difference equations in $ (n,x) $, using non-commutative Gröbner bases to eliminate variables and find annihilating operators.
• Proposes a hybrid continuous-discrete approach to make the problem tractable with current software, by combining recurrence relations and differential operators on the generating function.

Experimental results

Research questions

• RQ1 Can the q-TSPP conjecture be reduced to a finite number of routine computational checks using symbolic computation?
• RQ2 Is there a certificate function $ B(n,j) $ that satisfies the Soichi, normalization, and Okada conditions to verify the determinant identity?
• RQ3 Can the determinant evaluation be transformed into a constant-term identity in a generating function $ f_n(x) $, enabling algorithmic verification?
• RQ4 What is the most efficient computational strategy to verify the determinant identity using modern computer algebra systems?
• RQ5 Can the hybrid continuous-discrete method using differential-difference operators lead to a fully rigorous proof with current technology?

Key findings

• The q-TSPP conjecture is proven modulo a finite number of routine calculations, establishing the generating function $ \prod_{1\leq i\leq j\leq k\leq n} \frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}} $ for totally symmetric plane partitions.
• The determinant identity $ \det(a(i,j))_{1\leq i,j\leq n} = \prod_{1\leq i\leq j\leq k\leq n} \left( \frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}} \right)^2 $ is shown to be equivalent to the q-TSPP conjecture via Okada’s reduction.
• A certificate function $ B(n,j) $ is proposed that, if it satisfies the Soichi recurrence, normalization, and Okada evaluation, implies the correctness of the determinant identity.
• The problem is reformulated as constant-term identities in a generating function $ f_n(x) $, which allows for algorithmic verification using holonomic function theory.
• The continuous-discrete approach—using differential operators and recurrence relations—offers a feasible path to a fully rigorous proof with current software, though it remains computationally intensive.
• The authors conclude that while the proof is currently semi-rigorous, the probability of error is negligible compared to the risk of external existential doubt, justifying publication in a personal journal and on arXiv.