Is there a ''nice'' classification of **even** nilpotent elements in $\mathfrak{sl}\_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, such that for any (= every) $\mathfrak{sl}\_2$-triple $(e,f,h)$ in $\mathfrak{sl}\_n$, $ad(h)$ acts only with even eigenvalues.   

By ''nice'' classification I mean to have an algorithm which takes a partition an tells whether (the coresponding element) is even or not.



Characterisation of even nilpotent elements in $\mathfrak{sl}_n$
Per Collingwood-McGovern ([1993](https://mathscinet.ams.org/mathscinet-getitem?mr=1251060), Corollary 3.8.8) an element is even iff all labels on its weighted Dynkin diagram are 0 or 2. For partition $[d\_1,\dots,d\_k,0,\dots,0]$ these labels are the $h\_i-h\_{i+1}$, where $h\_1\geqslant h\_2\geqslant\dots\geqslant h\_n$ is a reordering of $\bigcup\_{i=1}^k\{d\_i-1, d\_i-3,\dots,-d\_i+1\}$ (*ibid.*, Lemma 3.6.4).


