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Mathilde Bouvel

Publications and source records attributed to Mathilde Bouvel.

At least 37 records · Page 2Linked to original sources
arXiv2019

Slicings of parallelogram polyominoes: Catalan, Schröder, Baxter, and other sequences

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 polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

math.CO↗
arXiv2018

Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1 \subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i$ which generalizes the one for the family $F_{i-1}$. The second topic of the paper is the enumeration of a fifth family $F_5$ of pattern-avoiding inversion sequences (containing $F_4$). This enumeration is also solved \emph{via} a succession rule, which however does not generalize the one for $F_4$. The associated enumeration sequence, which we call the \emph{powered Catalan numbers}, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted $Ω_{pCat}$ and $Ω_{steady}$, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the \emph{steady paths}, which are naturally associated with $Ω_{steady}$. They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with $Ω_{pCat}$). Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.

math.CO↗
arXiv2018

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 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.

math.PR↗
arXiv2018

Semi-Baxter and strong-Baxter: two relatives of the Baxter sequence

In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2-41-3$, which we call semi-Baxter permutations, and those avoiding the vincular patterns $2-41-3$, $3-14-2$ and $3-41-2$, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding $2-14-3$). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper. For each family (that of semi-Baxter -- or equivalently, plane -- and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite.

math.CO↗
arXiv2017

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 pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".

math.PR↗
arXiv2016

An algorithm computing combinatorial specifications of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.

math.CO↗
arXiv2016

Some families of trees arising in permutation analysis

We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these results to strong interval trees with a restriction on the arity of prime nodes. Doing so, we describe a filtration of the set of permutations based on their strong interval trees. This filtration is also studied from a purely analytical point of view, thus illustrating the convergence of analytic series towards a non-analytic limit at the level of the asymptotic behavior of their coefficients.

math.CO↗
arXiv2016

Analysis of Algorithms for Permutations Biased by Their Number of Records

The topic of the article is the parametric study of the complexity of algorithms on arrays of pairwise distinct integers. We introduce a model that takes into account the non-uniformness of data, which we call the Ewens-like distribution of parameter $θ$ for records on permutations: the weight $θ^r$ of a permutation depends on its number $r$ of records. We show that this model is meaningful for the notion of presortedness, while still being mathematically tractable. Our results describe the expected value of several classical permutation statistics in this model, and give the expected running time of three algorithms: the Insertion Sort, and two variants of the Min-Max search.

cs.DM↗
arXiv2014

An algorithm for deciding the finiteness of the number of simple permutations in permutation classes

In this article, we describe an algorithm to determine whether a permutation class C given by a finite basis B of excluded patterns contains a finite number of simple permutations. This is a continuation of the work initiated in [Brignall, Ruskuc, Vatter, Simple permutations: decidability and unavoidable substructures, 2008], and shares several aspects with it. Like in this article, the main difficulty is to decide whether C contains a finite number of proper pin-permutations, and this decision problem is solved using automata theory. Moreover, we use an encoding of proper pin-permutations by words over a finite alphabet, introduced by Brignall et al. However, unlike in their article, our construction of automata is fully algorithmic and efficient. It is based on the study of pin-permutations in [Bassino, Bouvel, Rossin, Enumeration of pin-permutations, 2011]. The complexity of the overall algorithm is O(n log n + s^{2k}) where n denotes the sum of the sizes of permutations in the basis B, s is the maximal size of a pin-permutation in B and k is the number of pin-permutations in B.

math.CO↗
arXiv2014

Permutation classes and polyomino classes with excluded submatrices

This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded submatrices) and we discuss how canonical such a description by submatrix-avoidance can be. We provide numerous examples of permutation and polyomino classes which may be defined and studied from the submatrix-avoidance point of view, and conclude with various directions for future research on this topic.

math.CO↗
arXiv2014

A general theory of Wilf-equivalence for Catalan structures

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences, or generating functions, of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal classes defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal classes have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of a given size and show that it is exponentially smaller than the corresponding Catalan number. In other words these "coincidental" equalities are in fact very common among principal classes. Our results also allow us to prove, in a unified and bijective manner, several known Wilf-equivalences from the literature.

math.CO↗
arXiv2014

Operators of equivalent sorting power and related Wilf-equivalences

We study sorting operators $\mathbf{A}$ on permutations that are obtained composing Knuth's stack sorting operator $\mathbf{S}$ and the reversal operator $\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we provide a size-preserving bijection between the set of permutations sorted by $\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ \mathbf{R} \circ \mathbf{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on a bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding pairs of Wilf-equivalent permutation classes.

math.CO↗
arXiv2013

Combinatorics of non-ambiguous trees

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.

math.CO↗
arXiv2012

Refined enumeration of permutations sorted with two stacks and a D_8-symmetry

We study permutations that are sorted by operators of the form $\mathbf{S} \circ α\circ \mathbf{S}$, where $\mathbf{S}$ is the usual stack sorting operator introduced by D. Knuth and $α$ is any $D_8$-symmetry obtained combining the classical reverse, complement and inverse operations. Such permutations can be characterized by excluded (generalized) patterns. Some conjectures about the enumeration of these permutations, refined with numerous classical statistics, have been proposed by A. Claesson, M. Dukes and E. Steingrímsson. We prove these conjectures, and enrich one of them with a few more statistics. The proofs mostly rely on generating trees techniques, and on a recent bijection of S. Giraudo between Baxter and twisted Baxter permutations.

math.CO↗
arXiv2012

Combinatorial specification of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positiveand algebraic. It also yields a uniform random sampler of permutations in C. The method presentedis fully algorithmic.

math.CO↗
arXiv2012

Geometric grid classes of permutations

A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.

math.CO↗
arXiv2012

Average-case analysis of perfect sorting by reversals (Journal Version)

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, polynomial, and additionally, for sufficiently long permutations, the sorting algorithm runs in polynomial time with probability one. Furthermore, our analysis of the subclass of commuting scenarios yields precise results on the average length of a reversal, and the average number of reversals.

cs.DM↗