Scaling limits of permutation classes with a finite specification: A dichotomy

  title={Scaling limits of permutation classes with a finite specification: A dichotomy},
  author={Fr{\'e}d{\'e}rique Bassino and Mathilde Bouvel and Valentin F{\'e}ray and Lucas Gerin and Mickael Maazoun and Adeline Pierrot},
  journal={Advances in Mathematics},

On random combinatorial structures: partitions, permutations and asymptotic normality

Random combinatorial structures form an active field of research at the interface between combinatorics and probability theory. From a theoretical point of view, some of the main objectives are to

The skew Brownian permuton: a new universality class for random constrained permutations

We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of various models of random constrained permutations. This family is parametrized by two real

Large deviation principle for random permutations

A large deviation principle for random permutations induced by probability measures of the unit square is derived and some properties of conditionally constant permutons are described with respect to inversions, which lead to a new notion ofpermutons, which generalizes both pattern-avoiding and pattern-packing permutations.

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random

Baxter permuton and Liouville quantum gravity

The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We find an explict formula for the expectation of the Baxter

Random cographs: Brownian graphon limit and asymptotic degree distribution

We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence toward a Brownian limiting object in the space of graphons. We then show that

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called \emph{tandem walks}, are well-known to be related to each other through several

Convergence law for $231$-avoiding permutations

. We prove that the class of 231-avoiding permutations satisfies a convergence law, i.e. that for any first-order sentence Ψ, in the language of two total orders, the probability p n, Ψ that a uniform

Locally uniform random permutations with large increasing subsequences

We investigate the maximal size of an increasing subset among points randomly sampled from certain probability densities. Kerov and Vershik’s celebrated result states that the largest increasing

A decorated tree approach to random permutations in substitution-closed classes

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from



Universal limits of substitution-closed permutation classes

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

On the growth of permutation classes

It is proved that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution, and a new enumeration technique is introduced, based on associating a graph with each permutation, and the generating functions for some previously unenumerated classes are determined.

Combinatoire et algorithmique dans les classes de permutations

An algorithm is given which derives a combinatorial specification for a permutation class given by its basis of excluded patterns, which is obtained if and only if the class contains a finite number of simple permutations, this condition being tested algorithmically.

Limits of permutation sequences

Simple permutations and pattern restricted permutations

The X-Class and Almost-Increasing Permutations

In this paper we give a bijection between the class of permutations that can be drawn on an X-shape and a certain set of permutations that appears in Knuth [4] in connection to sorting algorithms. A

The Brownian limit of separable permutations

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given

Enumeration of Pin-Permutations

A recursive characterization of the substitution decomposition trees of pin-permutations is given, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured in [18], the rationality of this generating function.

Formulae and Asymptotics for Coefficients of Algebraic Functions

When the function is a power series associated to a context-free grammar, the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers.