and $(2,3,5)$, which lead to the always-integral factorial ratio $$\begin{aligned}
v(n;(30,1),(2,3,5)) = \frac{{{\left\lfloor {\frac{n}{30}} \right\rfloor}}!\,\,\,n!}{{{\left\lfloor {\frac{n}{2}} \right\rfloor}}!{{\left\lfloor {\frac{n}{3}} \right\rfloor}}!{{\left\lfloor {\frac{n}{5}} \right\rfloor}}!},
\label{eqn.Chratio}\end{aligned}$$ thereby providing a positive example for Question \[q.alt\][^4]. Chebyshev used this ratio to prove the bound $$0.92 \frac{x}{\log x} \le \pi(x) \le 1.11 \frac{x}{\log x}$$ on the prime counting function, after careful analysis of the valuation of at all primes, which he knew was positive. Later, Diamond and Erdös [@DE] showed that one can use similar techniques to improve the constant in both of those bounds. It turns out that these improvements lead all the way to values of 1 for both constants, but proving this requires the Prime Number Theorem (which is of course equivalent to $\pi(x) \sim \frac{x}{\log x}$).

Historically, integrality information for ratios such as has been used to deliver elementary proofs of several statements in number theory. It is our hope that generalizations such as Question \[q.hook\] will eventually lead to more powerful techniques, exploiting the fact that partitions extend in 2 dimensions (this would give “more room” to construct interesting ratios).

The paper [@MR1784410] was partly motivated by similar questions: random matrix theory/number theory conjectures (the so-called Keating-Snaith conjectures) had previously led to 