Let $\lambda\vdash n$ denote a partition $\lambda$ of $n$ and let $\square\in\lambda$ denote a box $\square$ in the Young diagram of $\lambda$.
QUESTION. Can you list a pair of (distinct) statistics $(s(\square),t(\square))$ so that the following symmetry holds? $$\sum_{\lambda\vdash n}\sum_{\square\in\lambda}x^{s(\square)}y^{t(\square)} =\sum_{\lambda\vdash n}\sum_{\square\in\lambda}y^{s(\square)}x^{t(\square)}.\tag1$$
Example. If $s(\square)=h(\square)$ and $t(\square)=p(\square)$ stand for the hook-length and partition-length of a cell $\square$, then equation (1) holds (see this short paper by Bessenrodt and Han).
