Four factorization formulas for plane partitions

TL;DR

This paper introduces four identities linking all ten symmetry classes of plane partitions within a box, providing a unified approach to their enumeration and extending existing factorization theorems.

Contribution

It presents four new identities connecting symmetry classes of plane partitions and generalizes one of these identities in the style of a known factorization theorem.

Findings

Four identities connecting all ten symmetry classes of plane partitions.

A generalized identity extending previous factorization results.

Insights into the structure and enumeration of symmetric plane partitions.

Abstract

All ten symmetry classes of plane partitions that fit in a given box are known to be enumerated by simple product formulas, but there is still no unified proof for all of them. Progress towards this goal can be made by establishing identities connecting the various symmetry classes. We present in this paper four such identities, involving all ten symmetry classes. We discuss their proofs and generalizations. The main result of this paper is to give a generalization of one of them, in the style of the identity presented in "A factorization theorem for rhombus tilings," M. Ciucu and C. Krattenthaler, arXiv:1403.3323.