# Fully Complementary Higher Dimensional Partitions

## TL;DR

This paper introduces a new symmetry class for higher dimensional partitions and explores its properties and implications for plane partitions.

## Contribution

The paper introduces fully complementary higher dimensional partitions and proves a generating function for them.

## Key findings

- A formula for the generating function of fully complementary higher dimensional partitions is derived.
- New symmetry classes of plane partitions are conjectured and one is proven to be equinumerous to symmetric plane partitions.

## Abstract

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.