A Proof of George Andrews' and David Robbins' $q$-TSPP Conjecture
Bibliographic record
Abstract
Abstract. The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product-formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture. 1. Proemium In the historical conference Combinatoire Énumerative that took place at the end of May 1985, in Montreal, Richard Stanley raised some intriguing problems about the enumeration of plane partitions (see below), which he later expanded into a fascinating article [9]. Most of these problems concerned the enumeration of “symmetry classes ” of plane partitions that were discussed in more detail in another article of Stanley [10]. All of the conjectures in the latter article have since been proved (see David Bressoud’s modern classic [3]), except one, which until now 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, ca. 1983, independently by George Andrews and David Robbins ([10], [9] conj. 7, [3] conj. 13, and already alluded to in [1]). In the present article we finally turn this conjecture into a theorem. A plane partition pi is an array pi = (pii,j)1≤i,j, of positive integers pii,j with finite sum |pi | = pii,j, which is weakly decreasing in rows and columns so that pii,j ≥ pii+1,j and pii,j ≥ pii,j+1. A plane partition pi is identified with its 3D Ferrers diagram which is obtained by stacking pii,j unit cubes on top of the location (i, j). This gives a left-, back-, and bottom-justified structure in which we can refer to the locations (i, j, k) of the individual unit cubes. If the diagram is invariant under the action of the symmetric group S3 then pi is called a totally symmetric plane partition (TSPP). In other words, pi is called totally symmetric if whenever a location (i, j, k) in the diagram is occupied
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.026 | 0.004 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".