A proof of George Andrews’ and Dave Robbins’ q-TSPP conjecture (modulo a finite amount of routine calculations). The personal journal of Shalosh B. Ekhad and Doron Zeilberger

In the historic conference Combinatoire Énumérative [6] wonderfully organized by Gilbert La-belle 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 [7], 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 [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 Dave Bressoud’s modern classic [2]), 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 ([10], [9] (conj. 7), [2] (conj. 13)). In this tribute to Pierre Leroux, we describe how to prove that last stronghold. 1. q-TSPP: The Last Surviving Conjecture About Plane Partitions Recall that a plane partition pi is an array pi = (piij), i, j ≥ 1, of positive integers piij with finite sum |pi | = ∑piij, which is weakly decreasing in rows and columns. By stacking piij unit cubes on top of the ij location, one gets the 3D Ferrers diagram, that can be identified with the plane-partition, and is a left-, up-, and bottom- justified structure of unit cubes, and we can refer to the locations (i, j, k) of the individual unit cubes. A plane partition is totally symmetric iff whenever (i, j, k) is occupied (i.e. piij ≥ k), it follow