recalling parts of the theory for integral factorial ratios, based on the functions $f(x;\vec{\lambda},\vec{\gamma})$. In the balanced case, for integral factorial ratios of height $L-K$, Landau’s theorem and Bober’s theory lead to a $f(x;\vec{\gamma},\vec{\delta})$ with values in $\{0,1,\cdots,L-K\}$ (this is property **(P0)**). In addition, some extra properties are satisfied (these are easy to prove from the definition in ):

(P1)

:   $f$ is periodic, of period $P$ dividing $M=M(\vec{\gamma},\vec{\delta}) = \operatorname{lcm}(\gamma_1,\cdots,$ $\gamma_K,$ $\delta_1,\cdots,$ $\delta_L) $;

(P2)

:   for small enough $\epsilon>0$, we have $f(x)  + f(\epsilon+M-1-x) = L-K$ for all real $x$;

(P3)

:   $f(M-\epsilon) = f(P-\epsilon) = L-K$, for $\epsilon>0$ small enough.

Property **(P3)** is a direct consequence of **(P2)**, and **(P0)** is derived slightly more streinuously from **(P2)**.

By the discussion in Section \[sec.nonintegral\], for a given pair $(\vec{\gamma},\vec{\delta})$ to fail Question \[q.alt\], we need a $\mu$ such that $$\sum_k {{h}_{\gamma_k}\!\left(\mu\right)} - \sum_l {{h}_{\delta_l}\!\left(\mu\right)} < 0.\label{eqn.hookcount}$$ From this $\mu$, one can then construct (much bigger) examples of partitions $\lambda$ failing Question \[q.hook\] outright.

Since $h_i(\mu)$ counts hooks of $\mu$ divisible by $i$, an equivalent condition to is $$\sum_{\square \in \mu} g(h(\square))<0,
\label{eqn.equivalent}$$ if $g(x) = g(x;\vec{\gamma},\vec{\delta}):= \sum_k \mathbf{1}{(\gamma_k \text{ divides }x)} - \sum_l \mathbf{1}{(\delta_l \text{ divides }x)}$ (and 