of hook products.

Results
-------

We will actually proceed as for the classification of integral factorial ratios. This requires proving an analogue of Landau’s theorem, with adaptations to our setting.

\[thm.landaugeneral\] Let $(\gamma_k)$ and $(\delta_l)$ be two vectors of integral parameters. Then, the following two statements are equivalent: $$\frac{\prod_k {{{H}}_{\gamma_k}\!\left(\lambda\right)}}{\prod_l {{{H}}_{\delta_l}\!\left(\lambda\right)}} \text{ is integral for any partition $\lambda$;}\label{eqn.ratios}$$ and $$\sum_k {{h}_{\gamma_k}\!\left(\mu\right)} - \sum_l {{h}_{\delta_l}\!\left(\mu\right)} \ge 0 \text{ for any partition $\mu$.}
\label{eqn.counts}$$

If we require both $\lambda $ and $\mu$ to be one-rowed partitions, these two weaker statements are still equivalent and that result is precisely Landau’s original theorem. This theorem will be used to give easy proofs that would be very intricate otherwise, such as for the following theorem.

\[thm.multinomial\] Let $s$ and $t$ be positive integers. Then, $$\frac{{{{H}}_{s}\!\left(\lambda\right)}}{\left( {{{H}}_{st}\!\left(\lambda\right)}\right)^t}$$ is an integer for all partitions $\lambda$.

This theorem, when restricted to one-rowed partitions, of course reduces to the integrality of balanced multinomial coefficients.

Despite this good start, we are unable to give a complete classification of integral hook ratios. It is worth observing that the classification of integral factorial ratios is also incomplete.

For factorial ratios, in the special case of height 1, additional information is available: Bober 