Fully complementary higher dimensional partitions

TL;DR

This paper introduces a new class of higher-dimensional partitions called fully complementary partitions, provides a generating function formula, and explores their symmetry classes, leading to new conjectures and proofs in plane partition theory.

Contribution

It defines fully complementary higher dimensional partitions and derives their generating function, also studying their symmetry classes to propose and prove new conjectures.

Findings

Derived a formula for the generating function of FCPs

Defined new symmetry classes for plane partitions

Proved equinumerosity of quasi transpose complementary and 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.