Combinatorial meaning of the number of the even parts in a partition of $n$ into distinct parts

TL;DR

This paper provides a new direct combinatorial proof for the number of even parts in partitions of n into distinct parts, complementing previous generating function approaches and deepening understanding of partition combinatorics.

Contribution

It introduces a novel direct combinatorial proof for the enumeration of even parts in partitions into distinct parts, expanding on prior generating function methods.

Findings

New combinatorial proof established

Enhanced understanding of partition structure

Connections to Euler-Glaisher bijection clarified

Abstract

In a recent paper, Andrews and Merca investigated the number of even parts in all partitions of $n$ into distinct parts, which arise naturally from the Euler-Glaisher bijective proof. They obtained new combinatorial interpretations for this number by using generating functions. We obtain a new direct combinatorial proof in this note.