Valentin Féray
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View article: A Canonical Tree Decomposition for Order Types, and Some Applications
A Canonical Tree Decomposition for Order Types, and Some Applications Open
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as \emp…
View article: Tree-indexed sums of Catalan numbers
Tree-indexed sums of Catalan numbers Open
We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/π$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute the…
View article: A determinantal point process approach to scaling and local limits of random Young tableaux
A determinantal point process approach to scaling and local limits of random Young tableaux Open
International audience
View article: Binary search trees of permuton samples
Binary search trees of permuton samples Open
Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view…
View article: The permuton limit of random recursive separable permutations
The permuton limit of random recursive separable permutations Open
We introduce and study a simple Markovian model of random separable permutations. Our first main result is the almost sure convergence of these permutations towards a random limiting object in the sense of permutons, which we call the recu…
View article: A logical limit law for $231$-avoiding permutations
A logical limit law for $231$-avoiding permutations Open
We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permuta…
View article: Wiener Indices of Minuscule Lattices
Wiener Indices of Minuscule Lattices Open
The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. …
View article: Dense and nondense limits for uniform random intersection graphs
Dense and nondense limits for uniform random intersection graphs Open
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we …
View article: Asymptotic normality of pattern counts in conjugacy classes
Asymptotic normality of pattern counts in conjugacy classes Open
International audience
View article: A Canonical Tree Decomposition for Chirotopes
A Canonical Tree Decomposition for Chirotopes Open
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense …
View article: Asymptotic normality of pattern counts in conjugacy classes
Asymptotic normality of pattern counts in conjugacy classes Open
We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is alwa…
View article: A determinantal point process approach to scaling and local limits of random Young tableaux
A determinantal point process approach to scaling and local limits of random Young tableaux Open
We obtain scaling and local limit results for large random Young tableaux of fixed shape $λ^0$ via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: (1) an explicit descripti…
View article: Linear-sized independent sets in random cographs and increasing subsequences in separable permutations
Linear-sized independent sets in random cographs and increasing subsequences in separable permutations Open
This paper is interested in independent sets (or equivalently, cliques) in\nuniform random cographs. We also study their permutation analogs, namely,\nincreasing subsequences in uniform random separable permutations.\n First, we prove that…
View article: A logical limit law for $231$-avoiding permutations
A logical limit law for $231$-avoiding permutations Open
We prove that the class of 231-avoiding permutations satisfies a logical limit 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 random 231-avoiding permutation o…
View article: Scaling limit of graph classes through split decomposition
Scaling limit of graph classes through split decomposition Open
We prove that Aldous' Brownian CRT is the scaling limit, with respect to the Gromov--Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected distance-heredit…
View article: Components in Meandric Systems and the Infinite Noodle
Components in Meandric Systems and the Infinite Noodle Open
We investigate here the asymptotic behaviour of a large, typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system $\boldsymbol {M}_n$ on $2n$ points, as $n \rightarrow \infty $, to…
View article: Components in meandric systems and the infinite noodle
Components in meandric systems and the infinite noodle Open
We investigate here the asymptotic behaviour of a large typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system $M_n$ on $2n$ points, as $n \rightarrow \infty$, towards the infini…
View article: The interval posets of permutations seen from the decomposition tree perspective
The interval posets of permutations seen from the decomposition tree perspective Open
The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the …
View article: Random cographs: Brownian graphon limit and asymptotic degree distribution
Random cographs: Brownian graphon limit and asymptotic degree distribution Open
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 the degree of a uniform random vertex i…
View article: Random generation and scaling limits of fixed genus factorizations into transpositions
Random generation and scaling limits of fixed genus factorizations into transpositions Open
We study the asymptotic behaviour of random factorizations of the $n$-cycle into transpositions of fixed genus $g>0$. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was extens…
View article: Linear-sized independent sets in random cographs and increasing subsequences in separable permutations
Linear-sized independent sets in random cographs and increasing subsequences in separable permutations Open
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 separable permutations. First, we prove that, wi…
View article: Universal limits of substitution-closed permutation classes
Universal limits of substitution-closed permutation classes Open
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 suf…
View article: Cyclic inclusion-exclusion and the kernel of P -partitions
Cyclic inclusion-exclusion and the kernel of P -partitions Open
Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing f…
View article: Cumulants of Jack symmetric functions and b-conjecture (extended abstract)
Cumulants of Jack symmetric functions and b-conjecture (extended abstract) Open
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the f…
View article: Slicings of parallelogram polyominoes, or how Baxter and Schröder can be reconciled
Slicings of parallelogram polyominoes, or how Baxter and Schröder can be reconciled Open
We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram p…
View article: Central limit theorems for patterns in multiset permutations and set partitions
Central limit theorems for patterns in multiset permutations and set partitions Open
We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of …
View article: On the central limit theorem for the two-sided descent statistics in Coxeter groups
On the central limit theorem for the two-sided descent statistics in Coxeter groups Open
In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_n$ for which the two-sided descent statistics on a uniform random element of $W_n$ is asymptotically normal. Recently, Brück and Röttger …
View article: A decorated tree approach to random permutations in substitution-closed classes
A decorated tree approach to random permutations in substitution-closed classes Open
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 substitution-closed classes satisfying a critic…