Discrete Mathematics & Theoretical Computer Science
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. I... more Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.
It is known that binary words containing no k consecutive 1s are enumerated by k-step Fibonacci n... more It is known that binary words containing no k consecutive 1s are enumerated by k-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length n having this property.
For any pattern p ∈ {U,L,UU,UD,DU,DD}, we enumerate the equivalence classes of skew Dyck paths, w... more For any pattern p ∈ {U,L,UU,UD,DU,DD}, we enumerate the equivalence classes of skew Dyck paths, where two skew Dyck paths of the same semilength are p-equivalent whenever the positions of the occurrences of the pattern p are the same. In this paper we use generating functions, bijective arguments, and recurrence relations to obtain the main results.
For any pattern p of length at most two, we provide generating functions and asymptotic approxima... more For any pattern p of length at most two, we provide generating functions and asymptotic approximations for the number of pequivalence classes of Dyck paths with catastrophes, where two paths of the same length are p-equivalent whenever the positions of the occurrences of the pattern p are the same.
Discrete Mathematics & Theoretical Computer Science, 2017
We study the iteration of the process "a particle jumps to the right" in permutations. ... more We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length $n$ has typically $(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussia...
Recently, the authors introduced new families of Dyck paths having a first decomposition constrai... more Recently, the authors introduced new families of Dyck paths having a first decomposition constrained by the height or by the number of returns. In this work we extend the study to Motzkin paths and 2-colored Motzkin paths. For these new sets, we provide enumerative results by giving bivariate generating functions with respect to the length and another parameter, and we construct one-to-one correspondences with several restricted classes of ordered trees. We also deal with Schröder and Riordan paths. As a byproduct, we present a bijective proof of M(x)2 = 1 1−2xM( x 1−2x), where M(x) is the generating function of Motzkin numbers.
We study several restricted classes of compositions by giving one-to-one maps between them and di... more We study several restricted classes of compositions by giving one-to-one maps between them and different classes of restricted binary strings or pattern avoiding permutations. Inspired by the ECO method [8], new succession rules for these classes are presented. Finally, we obtain generating algorithms in Constant Amortized Time (CAT) for theses classes.
Genomes can be modeled by unsigned permutations of {1, 2, . . . , n} where each gene is assigned ... more Genomes can be modeled by unsigned permutations of {1, 2, . . . , n} where each gene is assigned to a unique number found once in the genome [6]. In this paper, we focus on the transposition mutation which consists in displacing an interval of the permutation. More precisely, our motivation is to find some combinatorial properties in terms of pattern avoiding permutations of the transposition mutation whenever we displace an interval of length one on its right, which is equivalent to delete one element and to insert it in a position located on its right. This operation will be called a deletion-right insertion (dri for short). Here is a dri-transformation on the permutation σ = 1 2 3 4 5 6 7:
Les arbres couvrants completement disjoints (CIST) presentent un reel interet dans les reseaux, a... more Les arbres couvrants completement disjoints (CIST) presentent un reel interet dans les reseaux, aussi bien pour des operations d'augmentation de robustesse que d'equilibrage de charge, de fractionnement du trafic, ... Leur etude theorique a montre de nombreux challenges lies a leur calcul et leur quantification. Nous proposons ici une formulation ILP originale et montrons par des resultats de simulation sur des modeles representatifs des reseaux sans fil que plusieurs CIST peuvent etre calcules lorsque la densite du reseau est suffisamment elevee. Nous montrons que dans ce type de reseaux, la densite et le nombre de noeuds sont proportionnels au nombre de CIST qui peuvent etre calcules.
A length-n Cayley permutation p of a total ordered set S is a length-n sequence of elements from ... more A length-n Cayley permutation p of a total ordered set S is a length-n sequence of elements from S, subject to the condition that if an element x appears in p then all elements y < x also appear in p. In this paper, we give a Gray code list for the set of length-n Cayley permutations. Two successive permutations in this list di er at most in two positions.
Sensor networks and Wireless Sensor Networks (WSN) are key components for the development of the ... more Sensor networks and Wireless Sensor Networks (WSN) are key components for the development of the Internet of Things. These networks are subject of two kinds of constraints. Adaptability by the mean of mutability and evolutivity, and constrained node resources such as energy consumption, computational complexity or memory usage. In this context, none of the existing protocols and models allows reliable peer authentication and trust level management. In the field of virtual economic transactions, Bitcoin has proposed a new decentralized and evolutive way to model and acknowledge trust and data validity in a peer network by the mean of the blockchain. We propose a new security model and its protocol based on the blockchain technology to ensure validity and integrity of cryptographic authentication data and associate peer trust level, from the beginning to the end of the sensor network lifetime.
For any patternof length at most two, we enumerate equivalence classes of Motzkin paths where two... more For any patternof length at most two, we enumerate equivalence classes of Motzkin paths where two paths of the same length are equivalent whenever they coin- cide on all occurrences of the pattern �.
Discrete Mathematics & Theoretical Computer Science
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. I... more Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.
It is known that binary words containing no k consecutive 1s are enumerated by k-step Fibonacci n... more It is known that binary words containing no k consecutive 1s are enumerated by k-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length n having this property.
For any pattern p ∈ {U,L,UU,UD,DU,DD}, we enumerate the equivalence classes of skew Dyck paths, w... more For any pattern p ∈ {U,L,UU,UD,DU,DD}, we enumerate the equivalence classes of skew Dyck paths, where two skew Dyck paths of the same semilength are p-equivalent whenever the positions of the occurrences of the pattern p are the same. In this paper we use generating functions, bijective arguments, and recurrence relations to obtain the main results.
For any pattern p of length at most two, we provide generating functions and asymptotic approxima... more For any pattern p of length at most two, we provide generating functions and asymptotic approximations for the number of pequivalence classes of Dyck paths with catastrophes, where two paths of the same length are p-equivalent whenever the positions of the occurrences of the pattern p are the same.
Discrete Mathematics & Theoretical Computer Science, 2017
We study the iteration of the process "a particle jumps to the right" in permutations. ... more We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length $n$ has typically $(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussia...
Recently, the authors introduced new families of Dyck paths having a first decomposition constrai... more Recently, the authors introduced new families of Dyck paths having a first decomposition constrained by the height or by the number of returns. In this work we extend the study to Motzkin paths and 2-colored Motzkin paths. For these new sets, we provide enumerative results by giving bivariate generating functions with respect to the length and another parameter, and we construct one-to-one correspondences with several restricted classes of ordered trees. We also deal with Schröder and Riordan paths. As a byproduct, we present a bijective proof of M(x)2 = 1 1−2xM( x 1−2x), where M(x) is the generating function of Motzkin numbers.
We study several restricted classes of compositions by giving one-to-one maps between them and di... more We study several restricted classes of compositions by giving one-to-one maps between them and different classes of restricted binary strings or pattern avoiding permutations. Inspired by the ECO method [8], new succession rules for these classes are presented. Finally, we obtain generating algorithms in Constant Amortized Time (CAT) for theses classes.
Genomes can be modeled by unsigned permutations of {1, 2, . . . , n} where each gene is assigned ... more Genomes can be modeled by unsigned permutations of {1, 2, . . . , n} where each gene is assigned to a unique number found once in the genome [6]. In this paper, we focus on the transposition mutation which consists in displacing an interval of the permutation. More precisely, our motivation is to find some combinatorial properties in terms of pattern avoiding permutations of the transposition mutation whenever we displace an interval of length one on its right, which is equivalent to delete one element and to insert it in a position located on its right. This operation will be called a deletion-right insertion (dri for short). Here is a dri-transformation on the permutation σ = 1 2 3 4 5 6 7:
Les arbres couvrants completement disjoints (CIST) presentent un reel interet dans les reseaux, a... more Les arbres couvrants completement disjoints (CIST) presentent un reel interet dans les reseaux, aussi bien pour des operations d'augmentation de robustesse que d'equilibrage de charge, de fractionnement du trafic, ... Leur etude theorique a montre de nombreux challenges lies a leur calcul et leur quantification. Nous proposons ici une formulation ILP originale et montrons par des resultats de simulation sur des modeles representatifs des reseaux sans fil que plusieurs CIST peuvent etre calcules lorsque la densite du reseau est suffisamment elevee. Nous montrons que dans ce type de reseaux, la densite et le nombre de noeuds sont proportionnels au nombre de CIST qui peuvent etre calcules.
A length-n Cayley permutation p of a total ordered set S is a length-n sequence of elements from ... more A length-n Cayley permutation p of a total ordered set S is a length-n sequence of elements from S, subject to the condition that if an element x appears in p then all elements y < x also appear in p. In this paper, we give a Gray code list for the set of length-n Cayley permutations. Two successive permutations in this list di er at most in two positions.
Sensor networks and Wireless Sensor Networks (WSN) are key components for the development of the ... more Sensor networks and Wireless Sensor Networks (WSN) are key components for the development of the Internet of Things. These networks are subject of two kinds of constraints. Adaptability by the mean of mutability and evolutivity, and constrained node resources such as energy consumption, computational complexity or memory usage. In this context, none of the existing protocols and models allows reliable peer authentication and trust level management. In the field of virtual economic transactions, Bitcoin has proposed a new decentralized and evolutive way to model and acknowledge trust and data validity in a peer network by the mean of the blockchain. We propose a new security model and its protocol based on the blockchain technology to ensure validity and integrity of cryptographic authentication data and associate peer trust level, from the beginning to the end of the sensor network lifetime.
For any patternof length at most two, we enumerate equivalence classes of Motzkin paths where two... more For any patternof length at most two, we enumerate equivalence classes of Motzkin paths where two paths of the same length are equivalent whenever they coin- cide on all occurrences of the pattern �.
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