Some hidden relations involving the ten symmetry classes of plane partitions

Abstract

Let B be a partially ordered product of three finite chains. For any group G of automorphisms of B, let N G ( B, q) denote the rank generating function for G-invariant order ideals of B. If we regard B as a rectangular prism, N G ( B, q) can be viewed as a generating function for plane partitions that fit inside B. Similarly, define N G ′( B, q) to be the rank generating function for order ideals of the quotient poset B G . We prove that N G ( B, − 1) and N G ′( B, − 1) count the number of plane partitions (i.e., order ideals of B) that are invariant under certain automorphisms and complementation operations on B. Consequently, one discovers that the number of plane partitions belonging to each of the ten symmetry classes identified by Stanley is of the form N G ( B, ± 1) or N G ′( B, ± 1) for some subgroup G of S 3, and conversely. We also discuss the occurrence of this phenomenon in general partially ordered sets, and use the theory of P-partitions to derive a criterion for one aspect of it.