Elementary Proofs of Infinitely Many Congruences for $k$-Elongated Partition Diamonds

TL;DR

This paper provides elementary proofs of infinitely many congruences for functions counting specific elongated partition diamonds, extending previous results with simpler methods.

Contribution

It introduces elementary proof techniques to establish infinitely many congruences for the partition functions $d_k$, expanding the known results beyond the cases previously proven using modular forms.

Findings

Proved infinitely many congruences for $d_k$ functions.

Extended results to an infinite set of $k$ values.

Used elementary $q$-series and generating function manipulations.

Abstract

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken $k$-diamond partitions. On the way to broken $k$-diamond partitions, Andrews and Paule introduced the idea of $k$-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of $k$-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers $d_k(n)$ produced by summing the links of $k$-elongated plane partition diamonds of length $n$ is given by $\frac{(q^2;q^2)_\infty^k}{(q;q)_\infty^{3k+1}}$ for each $k\geq 1.$ A significant portion of their recent paper involves proving several congruence properties satisfied by $d_1, d_2$ and $d_3$, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions $d_k$ for an infinite set of values of $k.$ The proof techniques employed are all elementary, relying on generating function manipulations and classical $q$-series results.