Robert Cori

• Université Bordeaux-I

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the r...

The aim of this paper is to come back to a data structure representation of graph by permutations. This originated in the years 1960-1970 by contributions due to J. Edmonds [7], A. Jacques [11], W. Tutte [22] in order to consider the embedding of a graph in a surface as a combinatorial object. Some algebraic developments where suggested in [4] and...

In this paper we consider doubly symmetric Dyck words, i.e. Dyck words which are fixed by two symmetries α and β introduced in [2]. We study combinatorial properties of doubly symmetric Dyck words, leading to the definition of two recursive algorithms to buisd these words. As a consequence we have a representation of doubly symmetric Dyck words as...

This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results. The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cor...

We define two new families of parking functions: one counted by Schröder numbers and the other by Baxter numbers. These families both include the well-known class of non-decreasing parking functions, which is counted by Catalan numbers and easily represented by Dyck paths, and they both are included in the class of underdiagonal sequences, which ar...

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our...

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our...

We define a permutation \(\varGamma _n\) on the set of words with n occurrences of the letter a and \(n+1\) occurrences of the letter b. The definition of this permutation is based on a factorization of these words that allows to associate a non crossing partition to them. We prove that all the cycles of this permutation are of odd lengths. We will...

Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transfor...

We examine three permutations on Dyck words. The first one, α, is related to the Baker and Norine theorem on graphs, the second one, β, is the symmetry, and the third one is the composition of these two. The first two permutations are involutions and it is not difficult to compute the number of their fixed points, while the third one has cycles of...

Indecomposable permutations in Sn+1, subgroups of index n of the free group on two generators and doubly pointed hypermaps of cardinality n are equinumerous. We give here a proof of a bijection, due to Sillke, between these three sets.

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavor. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configurations on graphs. The...

We define a solitary game, the Yamanouchi toppling game, on any connected graph of n vertices. The game arises from the well-known chip-firing game when the usual relation of equivalence defined on the set of all configurations is replaced by a suitable partial order. The set all firing sequences of length m that the player is allowed to perform in...

We study an extension of the chip-firing game. A given set of admissible moves, called Yamanouchi moves, allows the player to pass from a starting configuration � to a further configuration �. This can be encoded via an action of a certain group, the toppling group, associated with each connected graph. This action gives rise to a generalization of Ha...

International audience We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. Thi...

International audience We study an extension of the chip-firing game. A given set of admissible moves, called Yamanouchi moves, allows the player to pass from a starting configuration $\alpha$ to a further configuration $\beta$. This can be encoded via an action of a certain group, the toppling group, associated with each connected graph. This acti...

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavour. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configuration on graphs. The...

International audience We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distr...

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be...

We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed poi...

We give simple combinatorial proofs of some formulas for the number of factorizations of permutations in SnSn as a product of two nn-cycles, or of an nn-cycle and an (n−1)(n−1)-cycle.

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles...

The avalanche polynomial on a graph captures the distribution of avalanches in the abelian sandpile model. Studied on trees, this polynomial could be defined by simply considering the size of the subtrees of the original tree. In this article, we study some properties of this polynomial on plane trees. Previously it has been proved that two differe...

It is shown that the number of hypermaps of size n, that is the number of ordered pairs of permutations generating a transitive subgroup of Sn, is equal to (n−1)! times the number of indecomposable permutations of Sn+1. The proof is elementary.

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and...

International audience A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The er...

A permutation a 1a 2. . . a n is indecomposable if there does not exist p < n such that a 1a 2. . . a p is a permutation of f1, 2, . . . , pg. We compute the asymptotic probability that a permutation of Sn with m cycles is indecomposable as n goes to infinity with m/n fixed. The error term is O( log(n-m)/n-m ). The asymptotic probability is monoton...

We study the abelian sandpile model on different families of graphs. We introduced the avalanche polynomial which enumerates the size of the avalanches triggered by the addition of a particle on a recurrent configuration. This polynomial is calculated for several families of graphs. In the case of the complete graph, the result involves some known...

The assumption of the existence of global time, which significantly simplifies the analysis of distributed systems, is generally safe since most of the conclusions obtained under the global time axiom can be transferred to the frame where no such assumption is made. In this note, it was shown that the compositionality of the well-known correctness...

A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given. This mapping is such that the difference between the number of grains on a configuration and the external activity of the associate tree is the number of edges of the graph. It gives a bijective proof of a result of Merino López expressing...

In this paper we introduce and enumerate families of description trees. These families of trees consist of plane trees in which the nodes are labelled by nonnegative integers, and where the label of each node satisfies a condition relating it to the labels of its sons.We give a recursive construction of these trees which translates simply in an equ...

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p(1),..., p(k))-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some g...

A polynomial ideal encoding topplings in the abelian sandpile model on a graph is introduced. A Gröbner basis of this ideal is interpreted combinatorially in terms of well-connected subgraphs. This gives rise to algorithms to determine the identity and the operation in the group of recurrent configurations.

The group of recurrent configurations in the sandpile model, introduced by Dhar , may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The aim of this paper is to prove that the sandpile group of planar graph is isomorphic to that of its dual. A combinatorial point of view on the subject is al...

In this paper, we introduce description trees, to give a general framework for the recursive decompositions of several families of planar maps studied by W.T. Tutte. These trees reflect the combinatorial structure of the decompositions and carry out various combinatorial parameters. We also introduce left regular trees as canonical representants of...

We show how to express the sandpile model, introduced in theoretical physics, using the vocabulary of combinatorial theory. The group of recurrent configurations in the sandpile model, introduced by D. Dhar ([6]), may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The structure of the sandpi...

Abstract Dieren t formulas counting families of non isomorphic chord diagrams are given : planar and toroidal ones and those of maximal genus. These formulas are obtained establishing results on the structure of the automorphism,group of diagrams of a given genus. 3

Combinatorics on words, or finite sequences, is a field which grew simultaneously within disparate branches of mathematics such as group theory and probability. It has grown into an independent theory finding substantial applications in computer science automata theory and linguistics. This volume is the first to attempt to present a thorough treat...

The aim of this presentation is to give an introduction to the field of distributed algorithms in the shared memory model. An execution is considered as a set of elementary actions equiped with two relations satisfying some classical axioms. The notion of register is introduced and constructions of registers from elementary components is given. Thi...

Let the consensus number of a given set \(\mathcal{S}\) of shared objects, denoted \(\mathcal{C}\mathcal{N}(\mathcal{S})\), be the maximum number n such that there is a wait-free consensus protocol for n distinct processes, which communicate only by accessing objects in \(\mathcal{S}\). An interesting question concerning consensus numbers is the fo...

We consider the following problem: given three partitions A,B,C of a finite set Ω, do there exist two permutations α and β such that A,B,C are induced by α, β and αβ respectively? This problem is NP-complete. However it turns out that it can be solved by a polynomial time algorithm when some relations between the number of classes of A,B,C hold.

The aim of this paper is the study of asynchronous automata, a special kind of automata which encode the independency relation between actions and which enable their concurrent execution. These automata, introduced by Zielonka (RAIRO Inform. Theor. Appl.21, 99-135 (1987)), constitute a natural extension of finite automata to the case of asynchronou...

The aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2k - 1 and an effective construction has been given which needs k2k - 1 vertices....

We restrict the alphabet A to having three letters A={a,b,c} and the relation θ to have only one pair ac∼ θ ca. We prove that the number α n of words of length n in L 2 (A,θ) is bounded by a polynomial in n. This result is to be compared with the one obtained by Brandenburg wen θ=∅, proving that the number of square-free words of length n of {a,b,c...

On generalise la notion de mots sans carre et de mots sans carre abelien en introduisant celle de mot sans carre partiellement abelien pour une relation de commutation θ. Des resultats concernant le caractere fini ou infini de l'ensemble des mots sans carre partiellement abelien sont obtenus dans le cas des alphabets de trois ou quatre lettres

W. Zielonka has recently introduced a family of finite automata with a specific behavior, and called them asynchronous automata. They can be considered as a good model to describe concurrent processes exchanging data by means of some common storage. Hereafter, we restrict the family of asynchronous automata by defining the subclass of what we will...

The notion of trace was introduced in order to modelize the concurrency of actions. A trace is an element of the quotient of the free monoid by the congruence generated by a finite set of relations of the form abba. We introduce the notion of the approximation of a trace and we study its properties. The main result is the existence of an asynchrono...

The combinatorial investigation of graphs embedded on surfaces leads one to consider a pair of permutations (σ, α) that generate a transitive group [7]. The permutation α is a fixed-point-free involution and the pair is called a map . When this condition on α is dropped the combinatorial object that arises is called a hypermap . Both maps and hyper...

A formula for the number alternating Baxter permutations is given. The proof of this formula is given by constructing bijection between permutations, trees, and words. This gives also a combinatorial proof of a formula appearing in the enumerative theory of planar maps.

Without Abstract

We show that, in a free partially abelian monoid generated by a finite alphabet A, the subset [X∗] of A∗ containing all the words equivalent to a product of words of X is rational if X is a finite set of words, each word containing at least one occurrence of any letter of A. We suppose that the graph the vertices of which are letters of A and the e...

In this paper the following result is obtained: Let α and β be two permutations such that αβ is transitive and αp = βq = 1 (where p and q are distinct primes). Then the set of all permutations commuting both with α and β is either reduced to the identity or one of the three cyclic groups Cp, Cq or Cpq.

We show that given a finite group G a hypermap can be constructed having automorphism group isomorphic to G. The bipartite map of a modified version of this hypermap then affords a map with the same property.

In the theory of enumeration, the part devoted to the counting of planar maps gives rather surprising results. Of special interest to the combinatorialists is the conspicuous feature of counting numbers associated with families of maps as discussed in the papers of Tutte, Brown and Mullin. These formulas are by no means easy to prove; this is also...

The number of spanning trees in a graph is often called it's complexity [1]. A tree is of course of complexity one and it is a classical result of Cayley that the complete graph Kn has complexity nn-2. Between these two numbers lies the complexity of a connected graph.In the case of planar maps it is well-known that a map and its dual have equal co...

Les fonctions génératrices trouvées par W.T. Tutte et son école pour l'énumération des différents types de cartes planaires sont presque toujours des fonctions algébriques. Ici nous donnons l'énumération d'un type spécial de cartes planaires pour lesquelles les fonctions génératrices sont rationelles. La démonstration fait appel au code d'une carte...

L'énumération des grapises planaires a été developpée par W.T. Tutte et son école. Nous abordons le sujet par des moyens tout à fait différents' nous établissons une bijection entre les graphes planaires et un langage formel. Des relations vérifiées par ce langage formel dans l'algebre des sous-ensembles d'un monoide libre, nous déduisons des équat...

A bijection is defined between rooted planar maps, with a fixed number of vertices, and the words of a context-free language. Formal power series built from the previous language are shown to verify equations of a certain type, which are then solved, the results implying the algebraicity of the power series solution and giving enumerational formula...

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