The Catalan numbers are a very nice example for classical combinatorics - though maybe not the most famous one. But they deserve closer attention: Catalan numbers are really ubiquitous, they appear in various funny counting problems and are even known since Euler's times. We look at some of these counting problems. No prerequisites are needed for this part of the talk.
Towards the end of the talk, we discuss how Catalan numbers appear in the dimensions of certain endomorphism algebras - the combinatorics here are still very elementary, but some background knowledge of Lie algebras would not be amiss.