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

1 Proemium In the historical conference Combinatoire Énumerative that took place at the end of May 1985 in Montréal, Richard Stanley raised some intriguing problems about the enumeration of plane partitions (see below), which he later expanded into a fascinating article [11]. Most of these problems concerned the enumeration of “symmetry classes” of plane partitions that were discussed in more detail in another article of Stanley [12]. All of the conjectures in the latter article have since been proved (see David Bressoud’s modern classic [3]), except one, which until now has resisted the efforts of some 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 around 1983 independently by George Andrews and David Robbins ([12], [11] 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 π is an array π = (πi,j)1≤i,j , of nonnegative integers πi,j with finite sum |π| = ∑ πi,j , which is weakly decreasing in rows and columns so that πi,j ≥ πi+1,j and πi,j ≥ πi,j+1. A plane partition π is identified with its 3D Ferrers diagram which is obtained by stacking πi,j unit cubes on top of the location (i, j). The result is 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 on the coordinate axes then π is called a totally symmetric plane partition (TSPP). In other words, π is called totally symmetric if whenever a location (i, j, k) in the diagram is occupied then all its up to 5 permutations {(i, k, j), (j, i, k), (j, k, i), (k, i, j), (k, j, i)} are occupied as well. Such a set of cubes, i.e., all cubes to which a certain cube can be moved via S3 is called an orbit; the set of all orbits of π forms a partition of its diagram (see Figure 1). In 1995, John Stembridge [13] proved Ian Macdonald’s conjecture that the number of totally symmetric plane partitions with largest part at most n, i.e., those whose 3D Ferrers diagram is contained in the cube [0, n], is given by the elegant product-formula ∏