Week 27.04.2026 – 03.05.2026

In the area of combinatorics known as ‘permutation patterns’, seemingly innocuous questions can conceal a surprising degree of difficulty, giving rise to combinatorial problems that range from trivial to unsolved (despite decades of work). In this talk, we will explore several ways in which permutation pattern questions interface with probability — sometimes giving rise to probabilistic processes, sometimes illuminated by probabilistic reasoning. Some of these will be generalizations of known processes, others new, and still others conjectural.

Based on forthcoming joint work with Slim Kammoun and Einar Steingrimsson.

Consider the set of labelled trees on n vertices. Cayley’s formula tells us that there are n^{n-2} elements in this set; pick one uniformly at random. What can we say about this random tree in the limit as n goes to infinity? There are several different ways in which this (very vague!) question may be approached. One is via a scaling limit, that is, finding a good way to rescale the object with n (in this case, by giving each of its edges length n^{-1/2}) and then sending n to infinity, so that we obtain a continuum limit which is a random tree-like object. The limit in this case was discovered by David Aldous in 1990, and is known as the Brownian continuum random tree. My aim in this talk is to give an introduction to this beautiful area of probability theory.