The house of Hayashi was one of the Four houses.
The heads of the Hayashi Go School (林家) (1612-1884) were:
In 1884, Hayashi Shuei merged the School into the Honinbo School, becoming Honinbo Shuei.
All heads of the School from Gen'etsu through Hakuei were known when head as Hayashi Monnyu 林門入, in the same way as the standard Inoue Inseki and Yasui Senkaku. See iemoto system.
The Hayashi is the only School to never have a Meijin, if you discount Shuei.
Hayashi in Japanese means woods. It is said that when the Shogun offered a surname to this school, he suggested Mori 森, or forest. The recipient modestly declined and chose instead the more humble Hayashi.
John F. I think a better way of describing Shuei's translation back to the Honinbo family could be found, as the Hayashi family did not then cease. A collateral branch came down through Yubi - Hayashi Sano was the 16th generation, for example, which probably makes the Hayashi family the first to have a notable female player.