Enumeration of a symmetry class of plane partitions

Abstract

A cyclically symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box n =[1, n]×[1, n]×[1, n] and which is mapped to itself by cyclic permutations of the coordinate axes. Given a cyclically symmetric plane partition with Ferrers graph F, we can form its transpose-complement, the plane partition whose Ferrers graph is the set of all triples ( i, j, k) such that ( n + 1 − j, n + 1 − i, n + 1 − k) ∋ F. This is again a cyclically symmetric plane partition. A cyclically symmetric plane partition is tc-symmetric if it is equal to its transpose-complement. THis cannot occur unless n is even. In this paper we show that the number of tc-symmetric plane partitions in H 2n is given by π k=0 n−1 (3K+1)(6k)!(2k)! (4k+1)!(4k)! We show that, with a suitable assignment of weights, the generating function for tc-symmetric plane partitions divides the generating function for all cyclically symmetric plane partitions. We give analogous results for other classes of plane partitions including descending plane partitions.