the help of Figure \[fig.hooks\]). We get $30+0-15-10-6 = -1$ (a one-rowed $\mu$ would have given 0, see Theorem \[thm.bober\], page ). Now that we have found a $\mu$ that fails , we construct a $\lambda$ that fails . The largest hook length in $\mu$ is 10. We thus pick $p= 11$, and compute that the partition with empty core and quotients given by 11 copies of $\mu$ is $(66^{55})$, where evaluates to the *fraction* $$\frac{\substack{2^{60} 3^{53} 5^9 7^{35} 19^{12}  23^23  29^{29}  31^{31}  37^{37}  41^{41}  43^{43}  47^{47} 53^{53}  59^{55}  61^{55}  67^{54}  71^{50}  73^{48}  79^{42}\\  83^{38}  89^{32}  97^{24}  101^{20}  103^{18}  107^{14}  109^{12}  113^8}}{11^{11}}.$$ In order to show a set of parameters fails at Question \[q.hook\], it is thus efficient to exhibit a $\mu$ that fails Equation . For the classification at height 1, we will produce $\mu$s more systematically, of a very slim shape, and that are actually hooks themselves (*i.e.* their diagrams will not contain a $2\times 2$ rectangle).

Ratios of height 1
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To better appreciate Theorem \[thm.height1hook\], we first present the situations for integral factorial ratios. \[sec.height1\]

Classification of integral factorial ratios {#sec.factorialheight1}
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Based on an interpretation due to Rodriguez-Villegas of Question \[q.original\] in terms of hypergeometric 