Summer of Math Exposition

Nice strategy! It’s a good idea to try to motivate a lesser known construction with concrete examples and relations to a more well-known construction. But your current article follows a somewhat dry definition-theorem-proof style; I think it would be better if you lead with the problems involving Fuss-Catalan numbers, and introduce them through the problems. The equivalence between the recurrence and closed-form definitions is also not obvious, so you could spend more time deriving that.

Try to improve your prose; this is tricky, and the best way to do it is to read lots of math articles with an eye for how they write.

As a few comments:

• It’s good to avoid repetition — at the end of the proof of example 1.2, you start two sentences in a row with’ Thus’; varying your word choice can help your writing feel less monotonous.
• It’s good to use space to separate out ideas; at the end of your example 2.2, for instance, the “Note that…” comment could be given its own paragraph (or even remark environment).
• On that note, the construction “Note that” isn’t my favorite; I think most of the time you can just omit the note that and it reads better.

You also might want to look into the align* environment in LaTeX to make some of your equations look better.

It is a good article about generalizing Catalan number.

I think that this would be very helpful for students in a combinatorics class. The explanations are lucid and the proofs and diagrams are what I would expect from supplementary material for a course. However, I think that more review or path counting and other combinatorics results, even an informal one, might be helpful for people who have not encountered the subject material in a while and the piece could benefit from some more motivation.

It’s a paper. Well written, but indistinguishable from any other texts found in academia or education. In particular, it doesn’t have much of a motivation.

hmm