Enumeration modulo 4 of overpartitions wherein only even parts can be overlined

TL;DR

This paper provides a combinatorial proof and extension of a parity characterization for overpartitions with only even parts overlined, using classical and new involutions, and explores enumeration modulo 4.

Contribution

It introduces a combinatorial proof and extends the parity characterization of overpartitions to a modulus 4 setting, incorporating novel involution techniques.

Findings

Extended parity characterization to modulus 4

Developed combinatorial proofs using classical involutions

Provided generating function and combinatorial approaches

Abstract

In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be overlined. As part of that work, they used a generating function approach to prove a parity characterization for $d_2(n)$. In this note, we give a combinatorial proof of their result and extend it to a modulus 4 characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical involutions of Franklin, Glaisher, and Sylvester, along with a recent involution of van Leeuwen and methods new with this work.