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arXiv · 2411.03077

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

Aidan Carlson·Brian Hopkins·James A. Sellers
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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.

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Aidan Carlson, Brian Hopkins, James A. Sellers. 2024-11-05. Enumeration modulo 4 of overpartitions wherein only even parts can be overlined. https://doi.org/10.47443/dml.2024.130

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