Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution

Abstract: For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary $q$-series identities and generating function manipulations, along with mathematical induction: For all $\alpha \geq 1$ and all $n\geq 0,$ $$\mathrm{pend}\left(3^{2\alpha +1}n+\frac{17\cdot 3^{2\alpha}-1}{8}\right) \equiv 0 \pmod{3}, \textrm{ and}$$ $$\mathrm{pond}\left(3^{2\alpha +1}n+\frac{23\cdot 3^{2\alpha}+1}{8}\right) \equiv 0 \pmod{3},$$ where $\mathrm{pend}(n)$ counts the number of PEND partitions of weight $n$ and $\mathrm{pond}(n)$ counts the number of POND partitions of weight $n$. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin--Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.