Two formulas for plane paritions

TL;DR

This paper presents direct proofs of two identities related to plane partitions, one of which counts traces of plane partitions while the other lacks a combinatorial interpretation.

Contribution

It provides a new direct deduction and proof of two important identities in the theory of plane partitions, including one without a combinatorial interpretation.

Findings

Proof of an identity counting traces of plane partitions

Introduction of a second identity without combinatorial interpretation

Enhanced understanding of plane partition enumeration

Abstract

We give a direct deduction and proof of two identities in the theory of plane partitions. The first one is known to enumerate the traces of plane partitions. The second one comes without any combinatorial interpretation.