it is crucial to observe that ${{\mathcal{H}}_{r}\!\left({\mathbf{n}}\right)} = \left\{1,\cdots,\lfloor n/r\rfloor\right\}$, so ${{{H}}_{r}\!\left({\mathbf{n}}\right)} = \lfloor n/r \rfloor! $. In other words, if the partition is one-rowed, its hook product is a factorial that appears in Question \[q.alt\] (and similarly if it has only one column, of course). This suggests the following generalization, which will be primarily considered in this paper.

\[q.hook\] For which vectors of positive integers $\vec{\gamma} = (\gamma_k)$ and $\vec{\delta} = (\delta_l)$ is $$v(\lambda;\vec{\gamma},\vec{\delta}) := \frac{\prod_k{{{H}}_{\gamma_k}\!\left(\lambda\right)}}{\prod_l{{{H}}_{\delta_l}\!\left(\lambda\right)}}$$ an integer for all partitions $\lambda$?

It is immediately clear that the condition required by Question \[q.hook\] is stronger[^2] than the condition of Question \[q.alt\], since the latter corresponds to $\lambda = \mathbf{n}$ (*i.e.* one-rowed).

This accounts for some of the motivation of Question \[q.hook\]: Question \[q.alt\] admits only a partial answer at the moment, and one could thus hope that additional conditions the pairs $(\vec{\gamma},\vec{\delta})$ need to satisfy will slim the set to classify to a more manageable structure.

Motivation
----------

There is additional motivation for this new question, coming from both number theory and the representation theory of symmetric groups.

### Motivation in number theory

The problem posed originally by Question \[q.alt\] is connected to very deep questions in number theory, 