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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.

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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.

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