# Discrete mathematics & theoretical computer science DMTCS

Online ISSN: 1365-8050
Publications
Article
An {\it additive labeling} of a graph $G$ is a function $\ell :V(G) \o\mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ is joined to $y$). An {\it additive number} of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell :V(G) \to \lbrace 1,...,k\rbrace$. An {\it additive choosability number} of a graph $G$, denoted by $\eta_{\ell}(G)$, is the smallest number $k$ such that $G$ has an additive labeling from any assignment of lists of size $k$ to the vertices of $G$. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is $\mathbf{NP}$-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as planar triangle-free graphs and perfect graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\eta_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that contains a $(0,1)$-additive labeling, for all $\varepsilon >0$, approximating the $(0,1)$-additive number within $n^{1-\varepsilon}$ is $\mathbf{NP}$-hard.

Article
Rudolph conjectures that for permutations $p$ and $q$ of the same length, $A_n(p) \le A_n(q)$ for all $n$ if and only if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$, then $A_n(p) \le A_n(q)$ for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.

Article
International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

Article
International audience In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic $3D$-space. Here, we get such a cellular automaton with three states only.

Article
Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\geq (n\phi(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When $n$ is an odd prime we prove $s(n)\leq \frac{e^2}{\pi} n ((n-1)/e)^\frac{n-1}{2}$. For the second problem, we prove $2^{(n-1)/2}.(\frac{n-1}{2})!\leq t(n)\leq 2^k.(n-1)!/\phi(n)$ when $n$ is odd.

Article
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection} between Young tableaux and reduced webs. One main result uses Vogan's generalized $\tau$-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized $\tau$-invariants refine the data of the inversion set of a permutation. We define generalized $\tau$-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized $\tau$-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of the Robinson-Schensted correspondence. Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not $S_{3n}$-equivariant maps.

Article
We study Newton polytopes of cluster variables in type A_n cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type A cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph.

Article
International audience It is a well-known fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$-Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurement-based models of quantum computing.

Article
International audience We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients. Nous démontrons que les coefficients de Kronecker indexés par deux partitions de longueur au plus 2 sont donnés par des formules quasipolynomiales quadratiques dont les domaines de validité sont les cellules maximales d'un éventail. Des calculs simples nous donnent une description explicite des formules quasipolynomiales et de l'éventail associé. Ces nouvelles formulas sont obtenues de formules analogues pour les coefficients de Kronecker réduits correspondants et au moyen d'une formule reconstruisant les coefficients de Kronecker à partir des coefficients de Kronecker réduits. Une application est la caractérisation exacte de tous les coefficients de Kronecker non―nuls indexés par deux partitions de longueur au plus deux. Ceci nous a permis de réfuter une conjecture de Mulmuley au sujet des fonctions de dilatations associées aux coefficients de Kronecker.

Article
Using a recursive approach, we obtain a simple exact expression for the L^2-distance from the limit in R\'egnier's (1989) classical limit theorem for the number of key comparisons required by QuickSort. A previous study by Fill and Janson (2002) using a similar approach found that the d_2-distance is of order between n^{-1} log n and n^{-1/2}, and another by Neininger and Ruschendorf (2002) found that the Zolotarev zeta_3-distance is of exact order n^{-1} log n. Our expression reveals that the L^2-distance is asymptotically equivalent to (2 n^{-1} ln n)^{1/2}.

Article
Inspired by M. Haiman's Operator Theorem, we study $\frak{S}_n$-modules of polynomials in $\ell$ sets of $n$ variables, generated by a given homogeneous diagonally symmetric polynomial $f$. These modules are closed by taking partial derivatives, and generalized $\frak{S}_n$-invariants polarization operators. We completely classify these modules (according to Frobenius transform) when they are generated by degree 2 and degree 3 homogeneous symmetric polynomials. For the classification of modules associated to homogeneous degree 3 symmetric polynomials we introduce the notion of $n$-exception and we give an interesting conjecture to characterise this notion. We compute general formulas for the vector-graded Frobenius transform of $\frak{S}_n$-modules generated by degree 4 and degree 5 polynomials that seems to be universal.

Article
International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

Article
In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker alternately mark $a,b$ previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the $(a,b)$ game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all $(a,b)$ pairs for polyiamonds and polyominoes up to size four.

Article
International audience We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata. On donne une formule d’énumération exacte des automates finites déterministes acycliques minimaux. Cetteformule s’obtient à partir d’une bijection entre une famille fonctions de parking généralisées et les fonctions detransitions des automates acycliques.

Article
International audience Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.

Article
For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list edge $\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

Article
We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.

Article
Graphs and Algorithms International audience A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

Article
By the algorithm implemented in the paper  by Akiyama-Lee, we have examined pure discrete spectrum for some special cases of self-affine tilings.

Article
International audience We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations. Nous démontrons le $q$-analogue d'un résultat bien connu de Farahat et Higman : dans le centre de l'algèbre d'Iwahori-Hecke $\mathscr{H}_{n,q}$, si $(a_{\lambda \mu}^ν (n,q))_ν$ est l'ensemble des constantes de structure mises en jeu dans le produit de deux classes de conjugaison de Geck-Rouquier $\Gamma_{\lambda, n}$ et $\Gamma_{\mu,n}$, alors chaque coefficient $a_{\lambda \mu}^ν (n,q)$ dépend de façon polynomiale de $n$ et de $q$. Notre preuve repose sur la construction d'une limite projective des algèbres d'Hecke ; cette limite projective est inspirée de l'algèbre d'Ivanov-Kerov des permutations partielles.

Article
International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests. Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

Article
International audience We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.

Article
Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

Article
M.-P. Sch\"utzenberger asked to determine the support of the free Lie algebra ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We study these problems using the adjoint endomorphism $l^{*}$ of the left normed Lie bracketing $l$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. Calculating $l^{*}(w)$ via all factors of a given word $w$ of fixed length and the shuffle product, we recover the result of Duchamp and Thibon $(1989)$ for the support of the free Lie ring in a much more natural way. We rephrase these problems, for words of length $n$, in terms of the action of the left normed multi-linear Lie bracketing $l_{n}$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ - viewed as an element of the group ring of the symmetric group ${\mathcal S}_{n}$ - on $\lambda$-tabloids, where $\lambda$ is a partition of $n$. For words $w$ in two letters, represented by a subset $I$ of $[n] = \{1, 2, ..., n \}$, this leads us to the {\em Pascal descent polynomial} $p_{n}(I)$, a particular commutative multi-linear polynomial which equals to a signed binomial coefficient when $|I| = 1$ and allows us to obtain a sufficient condition on $n$ and $I$ in order that $w$ lies in ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. We also have a particular conjecture for twin and anti-twin words for the free Lie ring and show that it is enough to be checked for $|A| = 2$.

Article
We present here algebraic formulas associating a k-automaton to a k-epsilon-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-epsilon-automata into k-automata. The two automata have the same behaviour. Comment: 13 decembre 2004

Article
We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie combinatoire et l'alg\ebre homologique d'une classe de mono\"ides qui figurent naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interpr\'etation du nombre de Leray d'un complexe de clique en termes de la dimension globale d'une certaine alg\ebre non commutative.

Article
This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way.

Article
We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, Journal of Combinatorial Theory Series A, 110:237--273, 2005].

Article
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.

Article
Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of the previous chains. An important example is removing one "vertex" and reattaching it, in analogy with top-to-random shuffling. This larger family of Markov chains all admit analysis by Hopf-algebraic techniques. There are simple combinatorial expressions for their stationary distributions and for their eigenvalues and multiplicities and, in some cases, the eigenfunctions are also calculable.

Article
We show that the # product of binary trees introduced by Aval and Viennot [arXiv:0912.0798] is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.

Article
International audience We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411―430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type $B$. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781―2824]. Nous construisons des extensions unitaires des algèbres de pics d'ordre supérieur définies par Krob et le troisième auteur dans [Ann. Comb. 9 (2005), 411―430], et nous montrons qu'elles peuvent être obtenues comme images homomorphes de certaines sous-algèbres des algèbres de Mantaci-Reutenauer de type $B$. Ceci généralise un résultat dû à Bergeron, Nyman et au premier auteur [Trans. AMS 356 (2004), 2781―2824].

Article
International audience We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.

Article
International audience Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that \emphsurvey propagation (a certain message passing algorithm) is effective in finding proper k-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a k-coloring of graphs drawn from a certain planted-solution distribution over k-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem.

Article
A finite deterministic (semi)automaton $\mathcal{A} =(Q,\Sigma,\delta)$ is $k$-compressible if there is some word $w\in \Sigma^+$ such that the image of the state set $Q$ under the natural action of $w$ is reduced by at least $k$ states. Such word, if it exists, is called a $k$-compressing word for $\mathcal{A}$. A word is $k$-collapsing if it is $k$-compressing for each $k$-compressible automaton. We compute a set $\Gamma$ of short words such that all $3$-compressible automata on a two letter alphabet with a letter acting as a permutation are $3$-compressed by a word in $\Gamma$. Then we construct a shortest common superstring of the words in $\Gamma$ and we show that it is a $3$-collapsing word of length $53$. Moreover, as previously announced, we show that the shortest $3$-synchronizing word is not $3$-collapsing, illustrating the new bounds $34\leq c(2,3)\leq 53$ for the length $c(2,3)$ of the shortest $3$-collapsing word on a two letter alphabet.

Article
International audience I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.

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International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

Article
International audience We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ for primes $p \geq 5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark. Nous démontrons un $q$-analogue d'une congruence binomiale classique de Ljunggren qui stipule: $\binom{ap}{bp} \equiv \binom{a}{b}$ modulo $p^3$ pour $p$ premier tel que $p \geq 5$. Cette congruence s'inspire d'une précédente congruence prouvée par Babbage, Wolstenholme et Glaisher pour laquelle nous présentons les $q$-analogues existantes. Notre congruence généralise un précédent résultat de Clark.

Article
This paper develops an analytic theory for the study of some Polya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.

Article
International audience In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model. Dans un travail récent sur la physique statistique hors équilibre, un certain modèle d'exclusion Markovien appelé "processus d'annihilation asymétrique'' a été étudié par Ayyer et Mallick. Dans ce document, ils ont donné une conjecture précise pour les valeurs propres (avec les multiplicités) de la matrice stochastique. Ils ont en outre supposé que, pour chaque valeur propre, correspond un seul vecteur propre. Nous prouvons la première de ces conjectures en généralisant la matrice originale de Markov par l'introduction de paramètres supplémentaires, calculant explicitement ses valeurs propres, et en montrant que la nouvelle matrice se réduit à l'originale par une spécialisation appropriée. En outre, nous présentons un calcul de la fonction de partition dans le modèle généralisé, ce qui réduit également à celle obtenue par Ayyer et Mallick dans le modèle original.

Article
International audience We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials. Nous donnons deux interprétations combinatoires du Matrix Ansatz du PASEP en termes de chemins et de placements de tours. Cela donne deux preuves (presque) combinatoires d'une nouvelle formule pour la fonction de partition du PASEP. Cette formule donne aussi par exemple la fonction génératrice des permutations de taille donnée par rapport au nombre de montées et d'occurrences du motif $13-2$, la fonction génératrice par rapport au nombre d'excédences faibles et de croisements, et le $n^{\mathrm{ième}}$ moment de certains polynômes de $q$-Laguerre.

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International audience In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$. Nous étudions dans ce papier la distribution des distances dans les structures des réseaux apolloniens aléatoires (RANS), une famille de graphes en bijection avec les arbres ternaires planaires. En s'appuyant sur l'utilisation de séries génératrices multivariées pour décrire toute l'information sur les distances, ainsi que sur l'analyse de singularités pour évaluer les coefficients de ces séries, nous prouvons une distribution limite de Rayleigh pour les distances vers un sommet externe du RANS et montrons que la distance moyenne entre deux sommets quelconques d'un RANS d'ordre $n$ est asymptotiquement $\sqrt{n}$.

Article
Our aim is to construct a finite automaton recognizing the set of words that are at a bounded distance from some word of a given regular language. We define new regular operators, the similarity operators, based on a generalization of the notion of distance and we introduce the family of regular expressions extended to similarity operators, that we call AREs (Approximate Regular Expressions). We set formulae to compute the Brzozowski derivatives and the Antimirov derivatives of an ARE, which allows us to give a solution to the ARE membership problem and to provide the construction of two recognizers for the language denoted by an ARE. As far as we know, the family of approximative regular expressions is introduced for the first time in this paper. Classical approximate regular expression matching algorithms are approximate matching algorithms on regular expressions. Our approach is rather to process an exact matching on approximate regular expressions.

Article
Graphs and Algorithms International audience It is proved that there exist graphs of bounded degree with arbitrarily large queue-number. In particular, for all \Delta ≥ 3 and for all sufﬁciently large n, there is a simple \Delta-regular n-vertex graph with queue-number at least c√\Delta_n^{1/2-1/\Delta} for some absolute constant c.

Article
In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V \to W$ of sets of vertices such that $v \to w$ is an arc for every $v \in V$ and $w \in W$. An arc $v \to w$ is disimplicial when $N^-(w) \to N^+(v)$ is a diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.

Article
International audience We study the class of functions on the set of (generalized) Young diagrams arising as the number of embeddings of bipartite graphs. We give a criterion for checking when such a function is a polynomial function on Young diagrams (in the sense of Kerov and Olshanski) in terms of combinatorial properties of the corresponding bipartite graphs. Our method involves development of a differential calculus of functions on the set of generalized Young diagrams. Nous étudions la classe des fonctions sur l'ensemble des diagrammes de Young (généralisés) qui sont définies comme des nombres d'injections de graphes bipartites. Nous donnons un critère pour savoir si une telle fonction est une fonctions polynomiale sur les diagrammes de Young (au sens de Kerov et Olshanski) utilisant les propriétés combinatoires des graphes bipartites correspondants. Notre méthode repose sur le développement d'un calcul différentiel sur les fonctions sur les diagrammes de Young généralisés.

Article
International audience In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.

Article
Given an arrangement of lines in the plane, what is the minimum number $c$ of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between $\Omega (\log n / \log\log n)$. and $O(\sqrt{n})$. Similarly, we give bounds on the minimum size of a subset $S$ of the intersections of the lines in $\mathcal{A}$ such that every cell is bounded by at least one of the vertices in $S$. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $\Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $\Hcellzone$ hypergraph.

Article
The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number.

Article
Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $\det((a+j-i)\Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\Pf((j-i)\Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.