Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$

TL;DR

This work resolves Andrews and Paule's open problem by establishing infinite families of congruences for the partition-diamond function $PDN1(n)$ modulo powers of $5$ and $7$. It combines eta-quotient constructions with Atkin $U$-operators and higher-order modular equations (fifth and seventh order) to generate and manipulate four arithmetic-progressions of $PDN1(n)$, proving the congruences via modular-form arguments and Sturm-type bounds. The approach yields three families of congruences modulo $5^{\alpha}$ and two families modulo $7^{\alpha}$, mirroring classical Ramanujan and Watson-type results for $p(n)$ and extending the modular-equation toolkit to partition diamonds. These results enrich the theory of partition congruences and demonstrate a robust modular-forms framework for analyzing combinatorial partition statistics with special link-sum constraints.

Abstract

Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $ p(5^kn +d_k) \equiv 0 \pmod {5^k}$, where $24d_k\equiv 1 \pmod {5^k}$ and $k\geq 1$. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of $5$ for $PDN1(n)$. In addition, we prove two congruences modulo arbitrary powers of $7$ for $PDN1(n)$, which are analogous to Watson's congruences for $p(n)$.

Figures (2)

• Figure 1.1: A plane partition diamond of length $m$.
• Figure 1.2: Eighteen modified partition diamonds of 2 with $(n +1)$ copies of $n$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 2.6
• Theorem 3.1
• Lemma 3.2