Fully Complementary Higher Dimensional Partitions

We introduce a symmetry class for higher dimensional partitions—fully complementary higher dimensional partitions (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi-transpose-complementary plane partitions, are equinumerous to symmetric plane partitions.