Scaling limits of permutation classes with a finite specification: a dichotomy
Abstract
We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple permutations. Our goal is to study their limiting behavior in the sense of permutons. The limit depends on the structure of the specification restricted to families with the largest growth rate. When it is strongly connected, two cases occur. If the associated system of equations is linear, the limiting permuton is a deterministic $X$shape. Otherwise, the limiting permuton is the Brownian separable permuton, a random object that already appeared as the limit of most substitutionclosed permutation classes, among which the separable permutations. Moreover these results can be combined to study some non strongly connected cases. To prove our result, we use a characterization of the convergence of random permutons by the convergence of random subpermutations. Key steps are the combinatorial study, via substitution trees, of families of permutations with marked elements inducing a given pattern, and the singularity analysis of the corresponding generating functions.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.07522
 Bibcode:
 2019arXiv190307522B
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 60C05;
 05A05
 EPrint:
 72 pages, 27 figures