The Localization Method Applied to $k$-Elongated Plane Partitions and Divisibility by 5

TL;DR

This paper introduces a new localization method to prove an infinite family of divisibility congruences for the enumeration of 5-elongated plane partition diamonds, revealing novel algebraic structures with implications for partition theory.

Contribution

It develops a novel localization technique to establish divisibility properties of $k$-elongated plane partitions, a significant advancement over classical methods.

Findings

Discovered an infinite congruence family for $d_5(n)$ modulo powers of 5.

Uncovered a unique internal algebraic structure in the enumeration.

Demonstrated the effectiveness of the localization method in partition congruences.

Abstract

The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. We have discovered an infinite congruence family for $d_5(n)$ modulo powers of 5. Classical methods cannot be used to prove this family of congruences. Indeed, the proof employs the recently developed localization method, and utilizes a striking internal algebraic structure which has not yet been seen in the proof of any congruence family. We believe that this discovery poses important implications on future work in partition congruences.