Congruences for $k$-elongated plane partition diamonds

TL;DR

This paper proves and extends various congruences related to $k$-elongated plane partition diamonds, advancing understanding of their arithmetic properties and providing new families of congruences.

Contribution

It offers elementary proofs of existing conjectures, extends known congruences to new families, and discovers new congruences for $d_k(n)$ across multiple moduli.

Findings

Proved remaining conjectures of Andrews and Paule

Extended some congruences to new families

Found new congruences modulo various primes and powers

Abstract

In the eleventh paper in the series on MacMahons partition analysis, Andrews and Paule [1] introduced the $k$ elongated partition diamonds. Recently, they [2] revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the $k$ elongated plane partition diamonds of length $n$. Andrews and Paule [2] obtained several generating functions and congruences for $d_1(n)$, $d_2(n)$, and $d_3(n)$. They also posed some conjectures, among which the most difficult one was recently proved by Smoot [11]. Da Silva, Hirschhorn, and Sellers [5] further found many congruences modulo certain primes for $d_k(n)$ whereas Li and Yee [8] studied the combinatorics of Schmidt type partitions, which can be viewed as partition diamonds. In this article, we give elementary proofs of the remaining conjectures of Andrews and Paule [2], extend some individual congruences found by Andrews and Paule [2] and da Silva, Hirschhorn, and Sellers [5] to their respective families as well as find new families of congruences for $d_k(n)$, present a refinement in an existence result for congruences of $d_k(n)$ found by da Silva, Hirschhorn, and Sellers [5], and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64 and 128.