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Exactly how to think of fingerings and notations for the single B-flat horn is a point of confusion for many music educators. I believe this is a topic not covered well typically in a methods class.
A horn player reads from F horn music only, no matter what type of horn they play. If a person, perhaps a beginner, has a single B-flat horn really all this means is that they use B-flat horn fingerings but they still read from music in F like every one else. The fingerings for a single B-flat horn are exactly the same as the thumb down fingerings on a F/B-flat double horn. The fingerings for a single F horn are exactly the same as the thumb up fingerings on a F/B-flat double horn.
The important point to grasp is that horn players don’t look at B-flat horn fingerings like those of a trumpet or baritone–we think of them as being the B-flat side of a double horn fingerings. Pitches are still conceived of as being in F.
The confusion noted earlier is related to wondering the question can a player on a single B-flat horn read F horn music or do they need a different part? The answer is there is absolutely no problem with B-flat horn in relation to music in F. Most beginner methods present B-flat horn and F horn fingerings right there in the book and give both sets in the fingering chart. From the perspective of the students in a section where some players have single F and others single B-flat horns they just have different types of horns and they use somewhat different fingerings but they read from the same music and play at the same pitch level.
In terms of comparing the two types of single horns, the example I like to give my students is to just play them back to back, single F and single B-flat. The main thing to notice hearing the two is that the single B-flat horn sounds considerably better. If you were to play them back-to-back you would note that the single B-flat is easier to play also.
In short, there is no specific problem with fingerings or notations to overcome for the single B-flat horn. The player just uses the line of single B-flat horn fingerings, such as presented in my publication Introducing the Horn and pretty much every other introductory method out there for horn players.
One other footnote: notations and fingerings are the same for the B-flat marching horn. Parts should be notated in F and the fingerings are those of the B-flat side of a double horn, as described above.
