ensures the growth of $v(n;\vec{\gamma},\vec{\delta})$ is exponential rather than factorial in $n$. Historically, a closely related question is studied more often:

\[q.original\] For which vectors of positive integers $(\alpha_k : 1 \le k \le K)$ and $(\beta_l: 1 \le l \le L)$ is $$\label{eqn.original}
u(n;\vec{\alpha},\vec{\beta}) := \frac{\prod_{k=1}^K (\alpha_k n)!}{\prod_{l=1}^L(\beta_l n)!}$$ an integer for all integers $n$?

We will use footnotes to provide information in the context of Question \[q.original\], but these can be ignored for understanding our results[^1]: they are only there to provide links with existing literature.

Our main goal is to study a generalization of Question \[q.alt\], replacing $n$ with a partition.

Consider a partition $\lambda$, or rather its Young diagram. Let ${{\mathcal{H}}_{}\!\left(\lambda\right)}$ be the multiset of values of its hooks, and ${{{H}}_{}\!\left(\lambda\right)}$ the product of these values, sometimes called the hook product (see Section \[sec.def\] for definitions).

Let $${{\mathcal{H}}_{r}\!\left(\lambda\right)} := \left\{ \frac{h}{r}: h \in {{\mathcal{H}}_{}\!\left(\lambda\right)}| h \!\!\mod r = 0\right\},$$ (with repetitions) and similarly (with repetitions) ${{{H}}_{r}\!\left(\lambda\right)}$ the product of these values. We also set ${{h}_{r}\!\left(\lambda\right)}$ to be the cardinality of ${{\mathcal{H}}_{r}\!\left(\lambda\right)}$. We see that ${{\mathcal{H}}_{}\!\left(\lambda\right)} = {{\mathcal{H}}_{1}\!\left(\lambda\right)}$ and ${{{H}}_{}\!\left(\lambda\right)} = {{{H}}_{1}\!\left(\lambda\right)}$.

We will now use the notation ${\mathbf{n}}$ for the partition $(n)$. To generalize Question \[q.alt\], 