TITLE: Bijective proof for a partition identity
QUESTION [8 upvotes]: I came across the following cute fact about partitions:
\begin{align}
& |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt]
& {} - |\{ \lambda \vdash n \text{ with an odd number of even parts}\}| \\[8pt]
= {} & |\{ \lambda \vdash n \text{ which are self-conjugate  }  (\text{i.e. } \lambda = \lambda^\top )\}|
\end{align}
I have a simple proof via the representation theory of $S_n$ -- or really the result just fell out of a calculation I was doing. I was wondering if anyone knew a purely combinatorial bijective proof or had a reference for one.

REPLY [20 votes]: The answer is yes, there is a combinatorial proof, and both Sam's and Fёdor's proofs work.  However, this is a really old result and a combinatorial proof is old.  Here is the reference:
H. Gupta, Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Combin. Theory Ser. A, 21 (1976), no. 1, 100–103.
In general, you can find all of this in my survey.  This particular problem is §3.2.4.  This result goes back to Glaisher (1876), and the reference to Gupta is [79] given in the remarks at the end.  You would need to also use an obvious bijection in §2.1.5 which is the first step in Sam's answer.  According to Sylvester this is due to Durfee.