Wigner’s theorem

Abstract

We will encounter first the Catalan numbers as the number of (oriented) rooted trees. We shall define more precisely this object in the next paragraph. Actually, Catalan numbers count many other combinatorial objects. In a first part, we shall see that they also enumerate non-crossing partitions as well as Dick paths, a fact which we shall use later. As a warm-up to matrix models, we will also state the bijection with planar maps with one star. Then, we will study the Catalan numbers, their generating function, and relate them to the moments of the semicircle law.