Catalan words are particular growth-restricted words over the set of non-negative integers, and t... more Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern p we provide a bivariate generating function where the coefficient of x n y k in its series expansion is the number of length n p-avoiding Catalan words with k descents. As a byproduct, we enumerate the set of Catalan words avoiding p, and we provide the popularity of descents on this set.
Discrete Mathematics & Theoretical Computer Science, Jan 18, 2016
We explore the classical pattern avoidance question in the case of irreducible permutations, i.e.... more We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) -σ(i) = 1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length n -1 and the sets of irreducible permutations of length n (respectively fixed point free irreducible involutions of length 2n) avoiding a pattern α for α ∈ {132, 213, 321}. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
In this paper, we develop a constant amortized time (CAT) algorithm for generating permutations w... more In this paper, we develop a constant amortized time (CAT) algorithm for generating permutations with a fixed number of excedances. We obtain a Gray code for permutations having one excedance. We also give a bijection between the set of n-length permutations with exactly one excedance and the set Sn(321, 2413, 3412, 21534)\{123 . . . (n -1)n}. This induces a Gray code for the set S n (321, 2413, 3412, 21534)\{123 . . . (n -1)n}.
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.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2013
In , the author provided a Gray code for the set of n-length permutations with a given number of ... more In , the author provided a Gray code for the set of n-length permutations with a given number of left-to-right minima in inversion array representation. In this paper, we give the first Gray code for the set of n-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.
Les réseaux sans fils (WSN) souffrent aujourd'hui d'un manque de sécurité adaptée à leurs contrai... more Les réseaux sans fils (WSN) souffrent aujourd'hui d'un manque de sécurité adaptée à leurs contraintes multiples, auxquelles les solutions de gestion d'authentification et de confiance telles que PGP ne répondent que partiellement. D'une part, les contraintes d'autonomie et de coopération des noeuds nécessaires à la garantie de la cohésion du réseau nécessitent une solution distribuée, et d'autre part les contraintes de consommation énergétique et la faible puissance de calcul des noeuds imposent l'utilisation d'algorithmes de faible complexité [ZCW + 14]. À notre connaissance, aucune solution ne permet de répondre simultanément à ces deux problématiques. Nous proposons une nouvelle solution pour la sécurisation des WSNs nommée BATMAN(Blockchain Authentication and Trust Module in Ad-hoc Networks) qui répond à ces challenges. Nous présentons un modèle de gestion décentralisée pour l'authentification et la confiance, implementable sur la blockchain Tezos, et évaluons au travers de simulation les estimateurs de confiance proposés ici.
We give a Gray code and constant average time generating algorithm for derangements, i.e., permut... more We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no ÿxed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of ÿxed points bounded between two constants.
A pure excedance in a permutation π = π 1 π 2 . . . π n is a position i < π i such that there is ... more A pure excedance in a permutation π = π 1 π 2 . . . π n is a position i < π i such that there is no j < i with i ≤ π j < π i . We present a one-to-one correspondence on the symmetric group that transports pure excedances to descents of special kind. As a byproduct, we prove that the popularity of pure excedances equals those of pure descents on permutations, while their distributions are different.
Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pocke... more Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by allowing some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice paths in N 2 starting at the origin, made of steps U = (1, 1), D k = (1, −k), k ⩾ 1 and H = (1, 0), where two down steps cannot be consecutive, while the second one are lattice paths in N 2 starting at the origin, made of steps U , D k and H, where each step D k and H is necessarily followed by an up step, except for the last step of the path. We provide enumerative results for these paths according to the length, the type of the last step, and the height of its end-point. A similar study is made for these paths read from right to left. As a byproduct, we obtain new classes of paths counted by the Motzkin numbers. Finally, we express our results using Riordan arrays.
We provide generating functions for the popularity and the distribution of patterns of length at ... more We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
We give the first Gray code for the set of n-length permutations with a given number of cycles. I... more We give the first Gray code for the set of n-length permutations with a given number of cycles. In this code, each permutation is transformed into its successor by a product with a cycle of length three, which is optimal. If we represent each permutation by its transposition array then the obtained list still remains a Gray code and this allows us to construct a constant amortized time (CAT) algorithm for generating these codes. Also, Gray code and generating algorithm for n-length permutations with fixed number of left-to-right minima are discussed.
For any pattern α of length at most two, we enumerate equivalence classes of Lukasiewicz paths of... more For any pattern α of length at most two, we enumerate equivalence classes of Lukasiewicz paths of length n ≥ 0 where two paths are equivalent whenever the occurrence positions of α are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Lukasiewicz paths.
An n-multiset of [k] = {1, 2,. .. , k} consists of a set of n elements from [k] where each elemen... more An n-multiset of [k] = {1, 2,. .. , k} consists of a set of n elements from [k] where each element can be repeated. We present the bivariate generating function for n-multisets of [k] with no consecutive elements. For n = k, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern DU D, and we show how classical and novel statistics are transported by these bijections.
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value num... more Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the Möbius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give a bijective proof of Visontai's conjecture.
We study the distribution and the popularity of left children on sets of treeshelves avoiding a p... more We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.
Catalan words are particular growth-restricted words over the set of non-negative integers, and t... more Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern p we provide a bivariate generating function where the coefficient of x n y k in its series expansion is the number of length n p-avoiding Catalan words with k descents. As a byproduct, we enumerate the set of Catalan words avoiding p, and we provide the popularity of descents on this set.
Discrete Mathematics & Theoretical Computer Science, Jan 18, 2016
We explore the classical pattern avoidance question in the case of irreducible permutations, i.e.... more We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) -σ(i) = 1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length n -1 and the sets of irreducible permutations of length n (respectively fixed point free irreducible involutions of length 2n) avoiding a pattern α for α ∈ {132, 213, 321}. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
In this paper, we develop a constant amortized time (CAT) algorithm for generating permutations w... more In this paper, we develop a constant amortized time (CAT) algorithm for generating permutations with a fixed number of excedances. We obtain a Gray code for permutations having one excedance. We also give a bijection between the set of n-length permutations with exactly one excedance and the set Sn(321, 2413, 3412, 21534)\{123 . . . (n -1)n}. This induces a Gray code for the set S n (321, 2413, 3412, 21534)\{123 . . . (n -1)n}.
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.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2013
In , the author provided a Gray code for the set of n-length permutations with a given number of ... more In , the author provided a Gray code for the set of n-length permutations with a given number of left-to-right minima in inversion array representation. In this paper, we give the first Gray code for the set of n-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.
Les réseaux sans fils (WSN) souffrent aujourd'hui d'un manque de sécurité adaptée à leurs contrai... more Les réseaux sans fils (WSN) souffrent aujourd'hui d'un manque de sécurité adaptée à leurs contraintes multiples, auxquelles les solutions de gestion d'authentification et de confiance telles que PGP ne répondent que partiellement. D'une part, les contraintes d'autonomie et de coopération des noeuds nécessaires à la garantie de la cohésion du réseau nécessitent une solution distribuée, et d'autre part les contraintes de consommation énergétique et la faible puissance de calcul des noeuds imposent l'utilisation d'algorithmes de faible complexité [ZCW + 14]. À notre connaissance, aucune solution ne permet de répondre simultanément à ces deux problématiques. Nous proposons une nouvelle solution pour la sécurisation des WSNs nommée BATMAN(Blockchain Authentication and Trust Module in Ad-hoc Networks) qui répond à ces challenges. Nous présentons un modèle de gestion décentralisée pour l'authentification et la confiance, implementable sur la blockchain Tezos, et évaluons au travers de simulation les estimateurs de confiance proposés ici.
We give a Gray code and constant average time generating algorithm for derangements, i.e., permut... more We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no ÿxed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of ÿxed points bounded between two constants.
A pure excedance in a permutation π = π 1 π 2 . . . π n is a position i < π i such that there is ... more A pure excedance in a permutation π = π 1 π 2 . . . π n is a position i < π i such that there is no j < i with i ≤ π j < π i . We present a one-to-one correspondence on the symmetric group that transports pure excedances to descents of special kind. As a byproduct, we prove that the popularity of pure excedances equals those of pure descents on permutations, while their distributions are different.
Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pocke... more Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by allowing some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice paths in N 2 starting at the origin, made of steps U = (1, 1), D k = (1, −k), k ⩾ 1 and H = (1, 0), where two down steps cannot be consecutive, while the second one are lattice paths in N 2 starting at the origin, made of steps U , D k and H, where each step D k and H is necessarily followed by an up step, except for the last step of the path. We provide enumerative results for these paths according to the length, the type of the last step, and the height of its end-point. A similar study is made for these paths read from right to left. As a byproduct, we obtain new classes of paths counted by the Motzkin numbers. Finally, we express our results using Riordan arrays.
We provide generating functions for the popularity and the distribution of patterns of length at ... more We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
We give the first Gray code for the set of n-length permutations with a given number of cycles. I... more We give the first Gray code for the set of n-length permutations with a given number of cycles. In this code, each permutation is transformed into its successor by a product with a cycle of length three, which is optimal. If we represent each permutation by its transposition array then the obtained list still remains a Gray code and this allows us to construct a constant amortized time (CAT) algorithm for generating these codes. Also, Gray code and generating algorithm for n-length permutations with fixed number of left-to-right minima are discussed.
For any pattern α of length at most two, we enumerate equivalence classes of Lukasiewicz paths of... more For any pattern α of length at most two, we enumerate equivalence classes of Lukasiewicz paths of length n ≥ 0 where two paths are equivalent whenever the occurrence positions of α are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Lukasiewicz paths.
An n-multiset of [k] = {1, 2,. .. , k} consists of a set of n elements from [k] where each elemen... more An n-multiset of [k] = {1, 2,. .. , k} consists of a set of n elements from [k] where each element can be repeated. We present the bivariate generating function for n-multisets of [k] with no consecutive elements. For n = k, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern DU D, and we show how classical and novel statistics are transported by these bijections.
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value num... more Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the Möbius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give a bijective proof of Visontai's conjecture.
We study the distribution and the popularity of left children on sets of treeshelves avoiding a p... more We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.
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Papers by Jean-Luc Baril