This is a great article! Funny about Yugiri (39), because your version is a mirror image to Eawase (17), and the traditional Japanese Yugiri breaks that symmetry.
Every Genji-mon that is not self-symmetrical should have a mirror image, which I was able to quickly verify in the Genji-mon table you displayed.
Minori (40) is the only non symmetrical Genji-mon that breaks the rule, because it's the only one whose mirror image is also isomorphic to itself.
I can only imagine that there was something deliberate about the symmetry breaking for Yugiri, given the almost fanatical attention to detail in Japanese arts in general, and the equally strong penchant for deliberate imperfection in traditions like Kintsugi.
Loved this. Great writing and a bit of cultural arcana I knew nothing about but which resonates into the present day.
Obligatory entry in The On-Line Encyclopedia of Integer Sequences: https://oeis.org/A000110 which has 3 links mentioning Genji only one of which is not broken:
Xiaoling Dou, Hsien-Kuei Hwang and Chong-Yi Li, Bell numbers in Matsunaga's and Arima's Genjiko combinatorics [1]: Modern perspectives and local limit theorems, arXiv:2110.01156 [math.CO], 2021.
I wonder if a constraint like “prefer the single lines to be shorter” would favor the traditional Yūgiri over the current optimal version.
No irony detected despite the tyranny of low expectations for the ability of bored Japanese nobles.
A couple years ago I made a codepen to generate arbitrarily long Genji-mon from a ruleset, and managed to write a rule that reduces the number of required special cases to one. However, these rules are very fragile, and "break" with lengths > 5 (edit: or maybe they don't! this deserves further exploration). Glad to see someone else discovering the structure behind this interesting system!
https://codepen.io/rezmason/pen/ejWogV