@darijgrinberg Yes, my first thoughts were about the topics from Fulton's "Young tableaux with applications to...". But when I was reading this book by myself, its subject seemed to me very unnatural... Until the representation-theoretic applications arised. I'm afraid that maybe these children'll feel something like that.

@darijgrinberg Hello! When I was talking about the Okounkov-Vershik approach, I didn't mean that they know it. I only wanted to say that it's a quite famous topic... Well, if you want (do you?) to know more about the children mentioned in this question (and their background), I can write you an e-mail with more details (I don't think that I can write a lot of information about them here (or in other open places) without any permission -- nothing secret, of course, but I just think that it'll be impolite of me).

@MattF. I don't think that they know too much about manifolds: maybe some definitions and something like inverse function theorem. But I agree with you: their prerequisites're amazing.

A great material! Thank you!

@FrancoisZiegler I've edited my question especially for you.

@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...