A Proof of George Andrews’ and Dave Robbins’ q-TSPP Conjecture

In the historic conference Combinatoire Énumérative [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.