The subject of pattern-avoiding permutations is a classic of enumerative combinatorics, still rich of interesting open problems. We adopt a probabilistic point of view: What does the diagram of a large permutation in a pattern-avoiding class typically look like? Generalising previous results, we consider classes with nice encodings by multi-type trees. We show that they converge either to “Brownian separable permutons” or deterministic limit shapes.

I will explain how we use analytic combinatorics to study the scaling limit of the encoding trees without completely losing information about types and degrees of branch points.

If I have some time left, I will talk about some computations that we can perform on the limiting objects, with interesting consequences in the discrete.

This is joint work with F. Bassino, M. Bouvel, V. Féray, L. Gerin, A. Pierrot — arXiv:1903.07522.