Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice

TL;DR

This paper provides two elementary proofs for congruences related to overpartition functions where odd parts are repeated at most twice, addressing a problem posed by Merca and contributing to the understanding of partition congruences.

Contribution

It offers the first truly elementary proofs of specific congruences for a partition function with restrictions on odd parts, filling a gap in the literature.

Findings

Proved that a(4n+2) ≡ 0 mod 2

Proved that a(4n+3) ≡ 0 mod 2

Established elementary proofs for these partition congruences

Abstract

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all $n\geq 0$, \begin{align*} a(4n+2) &\equiv 0 \pmod{2}, \textrm{\ \ and} \\ a(4n+3) &\equiv 0 \pmod{2}. \end{align*} Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences.