The Enumeration of Totally Symmetric Plane Partitions

Abstract

A plane partition is totally symmetric if, as an order ideal of N3, it is invariant under all (six) permutations of the coordinate axes. We prove an explicit product formula for the number of totally symmetric plane partitions that fit in a cube of order n. This settles a conjecture published in 1980 by Andrews (who also attributed it to Macdonald and Stanley) and finishes the program of enumerating plane partitions belonging to each of the 10 symmetry classes proposed by Stanley in 1986. As a corollary of the proof, we also obtain a new proof of the formula, first obtained by Mills, Robbins, and Rumsey, for the number of plane partitions that are invariant under cyclic permutations and transposed complementation. © 1995 Academic Press. All rights reserved.