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 $PDN1\left(N\right)$ 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 $PDN1\left(n\right)$ and proved several congruences modulo 5,7,25,49 for $PDN1\left(n\right)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $p(5^{k}n+d_k)\equiv 0\left(\mathrm{mod}{5}^k\right)$, where $24d_k\equiv 1\left(\mathrm{mod}{5}^k\right)$ 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 $PDN1\left(n\right)$. In addition, we prove two congruences modulo arbitrary powers of 7 for $PDN1\left(n\right)$, which are analogous to Watson's congruences for $p\left(n\right)$.