Docentship Lecture

Title: Catalan combinatorics

Abstract: If you have never seen it, to continue the sequence 1,1,2,5,14,… in a logical way would be a difficult task. However, according to combinatorialist Richard Stanley, this sequence of Catalan numbers is “probably the most ubiquitous sequence of numbers in mathematics”. Algebraist Claus Michael Ringel claims it can be seen as the “heart of the theory of finite sets”, and the Online Encyclopedia of Integer Sequences lists the sequence as probably its longest entry, “and rightly so”.

Catalan numbers date back to the Chinese Mongolian mathematician Minggatu in the 1730s, and have been rediscovered several times. In this lecture we will discuss some of the many objects whose number is given by the Catalan numbers, among them possible orders of multiplications, binary trees, and triangulations of convex polygons. At the end, I will briefly sketch how this sequence is related current research in representation theory.

The lecture is an obligatory teaching test for those applying for admittance as docent and it should be possible for students and others with basic academic education in the relevant field to follow it. The lecture lasts 40-45 minutes with subsequent discussion. The lecture will be given in English.