A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method

TL;DR

This paper proves a family of congruences modulo powers of 3 for the 2-elongated plane partition function by expressing generating functions as modular functions within a specialized ring, advancing understanding of partition congruences.

Contribution

It introduces a novel modular functions approach to prove complex congruences for 2-elongated plane partitions, extending classical methods.

Findings

Proved an infinite family of congruences modulo powers of 3 for $d_2(n)$.

Expressed generating functions as elements of a ring of modular functions.

Applied localization techniques to establish the congruences.

Abstract

George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$.