A New Proof for a Triple Product Formula for Plane Partitions

TL;DR

This paper provides a new proof for Kamioka's finite analogue of Stanley's triple product formula for plane partitions with bounded parts, using tiling enumeration techniques.

Contribution

It introduces a novel proof method for Kamioka's formula, expanding the toolkit for analyzing plane partition generating functions.

Findings

New proof of Kamioka's finite analogue of Stanley's formula

Application of tiling enumeration techniques to plane partition identities

Enhanced understanding of bounded plane partition generating functions

Abstract

Stanley generalized MacMahon's classical theorem by proving a product formula for the norm-trace generating function for plane partition with unbounded parts. In his recent work on biothorgonal polynomials, Kamioka proved a finite analogue of Stanley's formula for plane partitions with bounded parts (arXiv:1508.01674). In this paper, we use techniques from the enumeration of tilings to give a new proof for Kamioka's formula.