A Proof of George Andrews' and David Robbins' $q$-TSPP Conjecture

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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