Universal limits of substitution-closed permutation classes

  • Frédérique Bassino

    Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France
  • Mathilde Bouvel

    Universität Zürich, Switzerland
  • Valentin Féray

    Universität Zürich, Switzerland
  • Lucas Gerin

    Ecole Polytechnique, Palaiseau, France
  • Mickaël Maazoun

    École Normale Supérieure de Lyon, France
  • Adeline Pierrot

    Université Paris-Sud, Orsay, France
Universal limits of substitution-closed permutation classes cover
Download PDF

A subscription is required to access this article.


We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons.

The limit depends on the generating series of the simple permutations in the class. Under a mild sufficient condition, the limit is an elementary one-parameter deformation of the limit of uniform separable permutations, previously identified as the Brownian separable permuton. This limiting object is therefore in some sense universal. We identify two other regimes with different limiting objects. The first one is degenerate; the second one is nontrivial and related to stable trees.

These results are obtained thanks to a characterization of the convergence of random permutons through the convergence of their expected pattern densities. The limit of expected pattern densities is then computed by using the substitution tree encoding of permutations and performing singularity analysis on the tree series.

Cite this article

Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, Mickaël Maazoun, Adeline Pierrot, Universal limits of substitution-closed permutation classes. J. Eur. Math. Soc. 22 (2020), no. 11, pp. 3565–3639

DOI 10.4171/JEMS/993