some unexpectedly-integral “bi-factorial ratios”, *i.e.* ratios of products of factorials at consecutive integers $\prod_{i=a}^b i$!. The authors studied these ratios in details in that paper, and their valuations at primes displayed intriguing fractal-like patterns. In the current paper, we are still interested in integer ratios, but considering a different construction.

In the context of using these integral hook ratios for number theory proofs, it is worth noting that an important component in the elementary proofs is the analytic continuation for $n!$ afforded by the $\Gamma$ function and the asymptotics given by the Stirling formula. Similarly, there exists an extensive theory of hook products, substituting for a discrete product over boxes of the diagram an integral over the shape of the diagram. This can also be replaced by an integral against the outline of the diagram (its graph after rotation of the diagram into the Russian orientation, given by a piecewise linear function of slope either equal to $\pm 1$ or not defined) of the Barnes $G$-function. For partitions of large size, this outline can be approximated by a Lipshitz 1 function (see [@KerovBook] for explanations along those lines). This can then be used to derive asymptotics as well as various 