Enumeration modulo four of overpartitions wherein only even parts may be overlined

In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers [Util. Math. 95 (2014) 33–49] defined d₂(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 d₂(n). In this article, we give a combinatorial proof of their result and extend it to a modulus four characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical bijections of Franklin, Glaisher, and Sylvester.