For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the [hook length][1] and [content][1] of $\square$, respectively.

R Stanley remarked [following Theorem 2.2, on page 6][2] that:
$$\sum_{n\geq0}x^n\sum_{\lambda\vdash n}\prod_{\square\in\lambda}\frac{(t+c_{\square})(v+c_{\square})}{h_{\square}^2}=(1-x)^{-tv}.$$
Something caught my attention:
>**QUESTION.** What is the conceptual or combinatorial reason that the right-hand side of
$$\sum_{n\geq0}x^n\sum_{\lambda\vdash n}\prod_{\square\in\lambda}\frac{a+\pmb{q}c_{\square}+c_{\square}^2}{h_{\square}^2}=(1-x)^{-a}$$
is independent of $\pmb{q}$?

[1]: http://www-math.mit.edu/~rstan/transparencies/hooks.pdf
[2]: http://www-math.mit.edu/~rstan/papers/hooks.pdf