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.

R Stanley remarked following Theorem 2.2, on page 6 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 $\pmb{q}$?

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.

R. Stanley remarked following Theorem 2.2, on page 6 of Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions 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 $\pmb{q}$?