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

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 $p(5^{k}n+d_k)\equiv 0(\mathrm{mod}{5}^k)$, where $24d_k\equiv 1(\mathrm{mod}{5}^k)$ and $k\ge 1$. 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 $p(n)$.