Elementary Proofs of Congruences for POND and PEND Partitions

TL;DR

This paper proves elementary Ramanujan-like congruences for POND and PEND partitions, which are variants of POD and PED partitions with specific restrictions on odd and even parts.

Contribution

It introduces new congruences for POND and PEND partitions and provides elementary proofs using classical q-series identities and induction.

Findings

Established infinite families of Ramanujan-like congruences modulo 3.

Provided elementary proofs relying on q-series identities.

Extended understanding of partition classes with odd/even part restrictions.

Abstract

Recently, Ballantine and Welch considered various generalizations and refinements of POD and PED partitions. 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). In the process, they were led to consider two classes of integer partitions which are, in some sense, the ``opposite'' of POD and PED partitions. They labeled these 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). In this work, we study these two types of partitions from an arithmetic perspective. Along the way, we are led to prove the following two infinite families of Ramanujan--like congruences: For all $\alpha \geq 1$ and all $n\geq 0,$ \begin{align*} pond\left(3^{2\alpha +1}n+\frac{23\cdot 3^{2\alpha}+1}{8}\right) &\equiv 0 \pmod{3}, \textrm{\ \ \ and} \\ pend\left(3^{2\alpha +1}n+\frac{17\cdot 3^{2\alpha}-1}{8}\right) &\equiv 0 \pmod{3} \end{align*} where $pond(n)$ counts the number of POND partitions of weight $n$ and $pend(n)$ counts the number of PEND partitions of weight $n$. All of the proof techniques used herein are elementary, relying on classical $q$-series identities and generating function manipulations, along with mathematical induction.