Jean-Luc Baril - Academia.edu (original) (raw)
Papers by Jean-Luc Baril
Aequationes mathematicae, Mar 6, 2024
Advances in Applied Mathematics, Aug 1, 2023
Discrete Mathematics & Theoretical Computer Science, Jan 18, 2016
HAL (Le Centre pour la Communication Scientifique Directe), 2009
HAL (Le Centre pour la Communication Scientifique Directe), 2022
HAL (Le Centre pour la Communication Scientifique Directe), Jun 30, 2020
International audienc
HAL (Le Centre pour la Communication Scientifique Directe), Oct 25, 2019
International audienc
arXiv (Cornell University), Jun 25, 2021
HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2013
Discrete Mathematics, Jun 1, 2023
arXiv (Cornell University), Apr 2, 2018
arXiv (Cornell University), Jan 6, 2021
Ars Mathematica Contemporanea
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.
arXiv (Cornell University), Nov 8, 2019
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.
Discrete Mathematics, Jun 1, 2007
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.
arXiv (Cornell University), Apr 4, 2018
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.
arXiv (Cornell University), Jun 27, 2019
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.
arXiv (Cornell University), Jun 25, 2016
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.
arXiv (Cornell University), Nov 23, 2016
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.
Aequationes mathematicae, Mar 6, 2024
Advances in Applied Mathematics, Aug 1, 2023
Discrete Mathematics & Theoretical Computer Science, Jan 18, 2016
HAL (Le Centre pour la Communication Scientifique Directe), 2009
HAL (Le Centre pour la Communication Scientifique Directe), 2022
HAL (Le Centre pour la Communication Scientifique Directe), Jun 30, 2020
International audienc
HAL (Le Centre pour la Communication Scientifique Directe), Oct 25, 2019
International audienc
arXiv (Cornell University), Jun 25, 2021
HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2013
Discrete Mathematics, Jun 1, 2023
arXiv (Cornell University), Apr 2, 2018
arXiv (Cornell University), Jan 6, 2021
Ars Mathematica Contemporanea
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.
arXiv (Cornell University), Nov 8, 2019
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.
Discrete Mathematics, Jun 1, 2007
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.
arXiv (Cornell University), Apr 4, 2018
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.
arXiv (Cornell University), Jun 27, 2019
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.
arXiv (Cornell University), Jun 25, 2016
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.
arXiv (Cornell University), Nov 23, 2016
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.