developed a complete classification in his thesis [@bober1]. We will present part of his work in Section \[sec.height1\]. The final result includes a relatively complex list of parameters, involving 3 infinite families and 52 sporadic cases (see Section \[sec.factorialheight1\]). In contrast, we present here the counterpart of this classification, but this time for hook ratios. The resulting list will reduce to essentially one case.

\[thm.height1hook\] Assume $\vec{\gamma} $ and $\vec{\delta}$ are such that $v(\lambda;\vec{\gamma},\vec{\delta})$ is integral for all $\lambda$, under the balancing condition that $\sum_k \frac{1}{\gamma_k} = \sum_l \frac{1}{\delta_l}$. Assume further that this hook ratio is of height 1, *i.e.* $L-K  = 1$.

Then, we actually have $K=1$ and thus $L=2$, and $2\gamma_1 = \delta_1 = \delta_2$.

Definitions {#sec.def}
===========

A *partition* $\lambda=(\lambda_1,\cdots,\lambda_d)$ is a weakly decreasing, finite sequence of positive integers, called its *parts*. The *size* of the partition $\lambda$, denoted $|\lambda|$, is the sum of its parts. We actually prefer to think of a partition in terms of its *Young diagram*. This allows to consider its *boxes*, and to define a *hook* based at the box $\square$ as the set of boxes exactly to the right, or exactly below $\square$, or $\square$ itself. This associates to each box 