New congruences on partition diamonds with $$n+1$$ copies of n

Abstract

Recently, Andrews and Paule introduced a partition function PDN1(N) which counts the number of partition diamonds with \(n+1\) copies of n where summing the parts at the links gives N. They also established the generating function of PDN1(n) and proved congruences modulo 5,7,25,49 for PDN1(n). At the end of their paper, Andrews and Paule asked for the existence of other types of congruence relations for PDN1(n). Motivated by their work, we prove some new congruences modulo 125 and 625 for PDN1(n) by using some identities due to Chern and Tang. In particular, we discover a family of strange congruences modulo 625 for PDN1(n). For example, we prove that for \(k\ge 0\),

$$\begin{aligned} PDN1\left( 5^7 \cdot 7^{8k}+\frac{ 19\cdot 5^7\cdot 7^{8k}+1 }{24} \right) \equiv 5^3 \pmod {5^4}. \end{aligned}$$