Some countability proofs

Theorem: The set of finite sequences of elements of a countable set is countable. I like this result because it specializes to several other basic countability results: for example, it implies that countable unions, finite products, and the set of finite subsets of countable sets are countable. I know several proofs of this result and [...]

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The Catalan numbers, regular languages, and orthogonal polynomials

I’ve been inspired by The Unapologetic Mathematician (and his pages and pages of archives!) to post more often, at least for the remainder of the summer. So here is a circle of ideas I’ve been playing with for some time. Let be the ordinary generating function for the ordered rooted trees on vertices (essentially we [...]

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