Je suis chargé de recherche CNRS à l'Institut Camille Jordan de l'Université Lyon 1, dans l'équipe de Combinatoire et Théorie des Nombres. Je co-organise le séminaire de l'équipe.

Je m'intéresse à des problèmes de combinatoire énumérative, bijective et algébrique.

## Preprints

1. On the length of fully commutative elements .
Description

In a Coxeter group W, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutation of adjacent letters. These elements have particularly nice combinatorial properties, and also index a basis of the generalized Temperley--Lieb algebra attached to W.
We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that it always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify those groups W for which the sequence is ultimately periodic, extending a result of Stembridge. These results are applied to the growth of generalized Temperley--Lieb algebras.

## Publications

1. Automata, reduced words, and Garside shadows in Coxeter groups ,
with Christophe Hohlweg and Nathan Williams. Journal of Algebra, volume 457 (2016), pp. 431-456.
Description

In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W,S). The definition of these automata is straightforward as it only requires the notion of weak order on (W,S) and the related notion of Garside shadows in (W,S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) the automata associated to the smallest Garside shadow is minimal; (2) the canonical automaton is minimal if and only if the support of all small roots is spherical, i.e., the corresponding root system is finite.

2. A Combinatorial Approach to a Model of Constrained Random Walkers ,
with Thibault Espinasse and Nadine Guillotin. Combinatorics, Probability and computing, volume 25, issue 02, pp. 222-235.
Description

In [1], the authors consider a random walk in Z K+1 with the constraint that each coordinate of the walk is at distance one from the following one. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with the one obtained in [1]

3. Long fully commutative elements in affine Coxeter groups ,
with Frédéric Jouhet. INTEGERS, Volume 15 (015), A36.
Description

An element of a Coxeter group W is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and affine Coxeter groups" (arXiv:1402.2166), R. Biagioli and the authors proved among other things that, for each irreducible affine Coxeter group, the sequence counting fully commutative elements with respect to length is ultimately periodic. In the present work, we study this sequence in its periodic part for each of these groups, and in particular we determine the minimal period. We also observe that in type A affine we get an instance of the cyclic sieving phenomenon.

4. Fully commutative elements in finite and affine Coxeter groups ,
with Riccardo Biagioli and Frédéric Jouhet. Monatshefte für Mathematik, Volume 178, Issue 1 (2015), Pages 1--37.
Description

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley--Lieb algebra. In this work we deal with any finite or affine Coxeter group W, and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley--Lieb algebra has linear growth.

An extended abstract was presented at Fpsac 2013 in Paris.
5. Combinatorics of fully commutative involutions in classical Coxeter groups ,
with Riccardo Biagioli and Frédéric Jouhet. Discrete Mathematics, Volume 338, Issue 12 (2015), Pages 2242--2259.
Description

An element of a Coxeter group WW is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot’s heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type AA, we connect our results to Fan–Green’s cell structure of the corresponding Temperley–Lieb algebra.

An extended abstract was presented at Fpsac 2013 in Paris.
6. Wieland gyration for triangular fully packed loop configurations,
with Sabine Beil and Ilse Fischer. Electronic Journal of Combinatorics, Volume 22, Issue 1 (2013), P26.
Description

Triangular fully packed loop configurations (TFPLs) emerged as auxiliary objects in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers Aπ of FPLs corresponding to a given link pattern π. The focus of this article is the definition and study of Wieland gyration on TFPLs. We show that the repeated application of this gyration eventually leads to a configuration that is left invariant. We also provide a characterization of such stable configurations. Finally we apply our gyration to the study of TFPL configurations, in particular giving new and simple proofs of several results.

7. Fully Packed Loop Configurations in a Triangle: matchings, paths and puzzles,
with Ilse Fischer.
Journal of Combinatorial Theory, series A, Volume 130 (2015), Pages 64-118.
Description

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. Firstly, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Secondly, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d(w)- d(u)-d(v)=0,1; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.

8. Tree-like tableaux
with Jean-Christophe Aval and Adrien Boussicault. Electronic Journal of Combinatorics, Volume 20, Issue 4 (2013), P34.
Description

In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tree-like tableaux of size n are counted by n! and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.

9. Fully Packed Loop Configurations in a Triangle
Journal of Combinatorial Theory, series A, Volume 120 (2013),Pages 2164-2188
Description

Fully Packed Loop configurations (FPLs) are certain configurations on the square grid, naturally refined according to certain link patterns. If $A_X$ is the number of FPLs with link pattern $X$, the Razumov--Stroganov correspondence provides relations between numbers $A_X$ relative to a given grid size. In another line of research, if $X\cup p$ denotes $X$ with $p$ additional nested arches, then $A_{X\cup p}$ was shown to be polynomial in $p$: the proof gives rise to certain configurations of FPLs in a triangle (TFPLs). In this work we investigate these TFPL configurations and their relation to FPLs. We prove certain properties of TFPLs, and enumerate them under special boundary conditions. From this study we deduce a class of linear relations, conjectured by Thapper, between quantities $A_X$ relative to different grid sizes, relations which thus differ from the Razumov--Stroganov ones.

10. Fully Packed Loop Configurations in a Triangle and Littlewood-Richardson coefficients
Journal of Combinatorial Theory, series A, Volume 120 (2013),Pages 2137-2147.
Description

In this work we continue our study of Fully Packed Loop (FPL) configurations in a triangle. These are certain subgraphs on a triangular subset of the square lattice, which first arose in the study of the usual FPL configurations on a square grid. We show that, in a special case, the enumeration of these FPLs in a triangle is given by Littlewood-Richardson coefficients. The proof consists of a bijection with Knutson-Tao puzzles.

11. Signed enumeration of ribbon tableaux: an approach through growth diagrams
with Dominique Gouyou-Beauchamps. Journal of Algebraic Combinatorics, Volume 36, Number 1 (2012), Pages 67-102.
Description

We give an extension of the famous Schensted correspondence to the case of ribbon tableaux, where ribbons are allowed to be of different sizes. This is done by extending Fomin’s growth diagram approach of the classical correspondence, in particular by allowing signs in the enumeration. As an application, we give in particular a combinatorial proof, based on the Murnaghan–Nakayama rule, for the evaluation of the column sums of the character table of the symmetric group.

12. The Structure of Alternative Tableaux
Journal of Combinatorial Theory, series A, Volume 118, Issue 5, July 2011, Pages 1638-1660.
arXiv version
Description

In this paper we study alternative tableaux introduced by Viennot. These tableaux are in simple bijection with permutation tableaux, defined previously by Postnikov. We exhibit a simple recursive structure for alternative tableaux. From this decomposition, we can easily deduce a number of enumerative results. We also give bijections between these tableaux and certain classes of labeled trees. Finally, we exhibit a bijection with permutations, and relate it to some other bijections that already appeared in the literature.

13. On some polynomials enumerating Fully Packed Loop configurations
with Tiago Fonseca. Advances in Applied Mathematics, Volume 47, Issue 3, September 2011, Pages 434-462.
arXiv version
Description

We are interested in the enumeration of Fully Packed Loop configurations on a grid with a given noncrossing matching. By the recently proved Razumov--Stroganov conjecture, these quantities also appear as groundstate components in the Completely Packed Loop model. When considering matchings with p nested arches, these numbers are known to be polynomials in p. In this article, we present several conjectures about these polynomials: in particular, we describe all real roots, certain values of these polynomials, and conjecture that the coefficients are positive. The conjectures, which are of a combinatorial nature, are supported by strong numerical evidence and the proofs of several special cases. We also give a version of the conjectures when an extra parameter tau is added to the equations defining the groundstate of the Completely Packed Loop model.

14. A Bijection Between Well-Labelled Positive Paths and Matchings
with Olivier Bernardi and Bertrand Duplantier, Séminaire Lotharingien de Combinatoire (2010), volume 63, Article B63e.
Description

A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p1p2...pn-1 on the alphabet {-1,0,+1} such that \sum_{i=1}^j pi >= 0 for all j=1,...,n-1, together with a permutation \sigma=\sigma1\sigma2...\sigman of {1,...,n} such that pi=-1 implies \sigmai<\sigmai+1, while pi=1 implies \sigmai> \sigmai+1. We establish a bijection between well-labelled positive paths of size n and matchings (i.e., fixed-point free involutions) on {1,...,2n}. This proves that the number of well-labelled positive paths is (2n-1)!!=(2n-1).(2n-3)...3.1.

Well-labelled positive paths appeared recently in the author's article "Partition function of a freely-jointed chain in a half-space" [in preparation] as a useful tool for studying a polytope \Pin related to the space of configurations of the freely-jointed chain (of length n) in a half-space. The polytope \Pin consists of points (x1,...,xn) in [-1,1]n such that \sum_{i=1}^j xi >= 0 for all j=1,...,n, and it was shown that well-labelled positive paths of size n are in bijection with a collection of subpolytopes partitioning \Pin. Given that the volume of each subpolytope is 1/n!, our results prove combinatorially that the volume of \Pin is (2n-1)!!/n!.

Our bijection has other enumerative corollaries in terms of up-down sequences of permutations. Indeed, by specialising our bijection, we prove that the number of permutations of size n such that each prefix has no more ascents than descents is [(n-1)!!]2 if n is even and n!!(n-2)!! if n is odd.

15. Growth function for a class of monoids
with Marie Albenque, FPSAC 2009, Linz.
Description

In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra.

These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras.

16. Comparing State Spaces in Automatic Security Protocol Verification
with Cas Cremers and Pascal Lafourcade, Lecture Notes in Computer Science 5458, pages 70-94.
Description

There are several automatic tools available for the symbolic analysis of security protocols. The models underlying these tools differ in many aspects. Some of the differences have already been formally related to each other in the literature, such as difference in protocol execution models or definitions of security properties. However, there is an important difference between analysis tools that has not been investigated in depth before: the explored state space. Some tools explore all possible behaviors, whereas others explore strict subsets, often by using so-called scenarios.

We identify several types of state space explored by protocol analysis tools, and relate them to each other. We find previously unreported differences between the various approaches. Using combinatorial results, we determine the requirements for emulating one type of state space by combinations of another type.

We apply our study of state space relations in a performance comparison of several well-known automatic tools for security protocol analysis. We model a set of protocols and their properties as homogeneously as possible for each tool. We analyze the performance of the tools over comparable state spaces. This work enables us to effectively compare these automatic tools, i.e., using the same protocol description and exploring the same state space. We also propose some explanations for our experimental results, leading to a better understanding of the tools

Remark: as this abstract suggests, the results of this article are quite far from my usual research themes. My contributions to this article are a nice application of Pólya-Redfield enumeration, and some necessary effort to understand the work of my coauthors.

17. A normalization formula for the Jack polynomials in superspace and an identity on partitions
with Luc Lapointe and Yvan Le Borgne, Electronic Journal of Combinatorics, volume 16(1), Article #R70.
Description

We prove a conjecture of Desrosiers, Lapointe and Mathieu [Adv. Math. 212, 2007, p.361--388] giving a closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.

18. Bijections for permutation tableaux
with Sylvie Corteel, European Journal of Combinatorics, volume 30(1), 2009, pages 295--310.
Description

In this paper we propose two bijections between permutation tableaux and permutations. These bijections show how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use those bijections to define subclasses of permutation tableaux that are in bijection with set partitions.

19. Growth diagrams, Ribbon tableaux and the Involution principle
with Dominique Gouyou-Beauchamps, FPSAC 2007, Tianjin.
Description

Sergey Fomin defined a very general setting of dual graded graphs that extends the classical Schensted correspondence to a much wider class of objects. His approach is bijective and also has an algebraic side inspired by the works of Stanley on differential posets. In this work we present a possible extension of Fomin's work through the signed enumeration of ribbon tableaux, where ribbons are not constrained.

20. A General Bijection for Walks on the Slit Plane
FPSAC 2006, San Diego.
Description

We study walks in the plane $\mathbb{Z}^2$, with steps in a given finite set $\mathcal{S}$, which start from the origin but otherwise never hit the half-line $\mathcal{H}=\{(k,0),k\leq 0\}$. These walks on the slit plane have received some attention these last few years, since in particular their enumeration leads to simple closed formulas; but only one bijection has been found so far, in the case of the square lattice, and in this work we give bijections extending this to other set steps.

Let $p=p(S)$ be the smallest possible abscissa x such that there is a walk on the slit plane ending at $(x,0)$. Suppose that $|j|\leq 1$ for each step $(i,j)\in\mathcal{S}$. The main result of this paper is the construction of a length preserving bijection between S-walks on the slit plane with a marked step ending at $(p,0)$, and a certain class of walks on the plane whose enumeration is much simpler. This allows us to interpret combinatorially previously known enumerations, and to give many new ones.

21. Enumeration of Walks Reaching a Line
EUROCOMB 2005.
Description

We enumerate walks in the plane $\mathbb{R}^2$, with steps East and North, that stop as soon as they reach a given line; these walks are counted according to the distance of the line to the origin, and we study the asymptotic behavior when the line has a fixed slope and moves away from the origin. When the line has a rational slope, we study a more general class of walks, and give exact as well as asymptotic enumerative results; for this, we define a nice bijection from our walks to words of a rational language. For a general slope, asymptotic results are obtained; in this case, the method employed leads us to find asymptotic results for a wider class of walks in $\mathbb{R}^m$.

22. On the Number of Fully Packed Loop Configurations with a Fixed Associated Matching
with Fabrizio Caselli, Bodo Lass, and Christian Krattenthaler, Electronic Journal of Combinatorics, volume 11(2), Article #R16, 41 pp.
Description

We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11 (2004), Article #R13].

## In preparation

1. Fully commutative elements and rationality.

## Exposés

Plus ou moins récents.