=  15+24 \in A_0+A_0$. Therefore, we have found an example partition $\mu$ that does not satisfy : it has an arm length $a=15$, and leg length $l=24$, *i.e.* $(16,1,\cdots,1)$, with 24 trailing zeroes. This $\mu$ can be used to construct a $\lambda$ that fails to satisfy , using the technique of Section \[sec.nonintegral\].

Bounds
======

For integral factorial ratios, some bounds are known [@bober2] for $K+L$ given the value of $L-K$. These techniques were improved in [@Schmerling]. The bounds obtained can be stated in different ways: as an explicit bound on $K+L$ as a function of $L-K$, as an asymptotic bound on the growth of that function, or optimized and explicit for specific values of $L-K$.

For the small values of $L-K$, these bounds seem to overestimate the actual value of $L+K$: in the case of $L-K=1$ for instance, the exhaustive classification of Bober (Theorem \[thm.bober\]) gives a maximal value of 9 for $K+L$ (for one of the 52 sporadic cases). Schmerling’s optimized version of the bound when $L-K=1$ gives an upper bound of 43, still far off the truth. For $L-K=2$ or 3, non-exhaustive computer searches point to a similar overshoot of the upper bounds.

As Question \[q.hook\] is 