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This is a 2nd installment to my earlier MO question for which Mark Wildon furnished a clean answer.

$\mathcal{O}(\pi)$ and $\mathcal{E}(\pi)$ stand for the number of odd and even cycles of a permutation $\pi$, respectively.

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.

Question. Is there a similar two-variable (generalized) formula to express $$\frac1{n!}\sum_{\pi\in\mathfrak{S}_n}t^{\mathcal{O}(\pi)}q^{\mathcal{E}(\pi)}$$ in terms of hook-length $h_{\square}$ and contents $c_{\square}\,$?

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