Number of even parts of a partition Fix a positive integer $n$. For a partition $\lambda$ of $n$, let $e(\lambda)$ be the number of even parts in $\lambda$. Using generating functions or bijections, we can show the statistic $e(\lambda)$ has the same distribution on the set of partitions of $n$ as the following statistics:
(1) The largest part in $\lambda$ that occurs more than once.
(2) $\sum \lfloor a_i/2\rfloor$, where $a_i$ is the number of occurrences of $i$ in $\lambda$.
(3) The number of cells in the Young diagram of $\lambda$ whose leg is zero and arm is odd.
Question: are there any other statistics that have the same distribution as $e(\lambda)$? In particular, I'm interested in something that is more like (3) and does not involve divisibility test.

Well, I have seen $e(\lambda)$ and (2), but not (1) and (3).  These statistics come up in Andrews's identity generalized by Cilanne Boulet here.  I suggest exploring that paper, the symmetries of the 4-variable g.f. and perhaps generalizing it, as (1) would suggests.  See my survey for the background.  
