0000000000288877
AUTHOR
Carine Khalil
Catalan and Schröder permutations sortable by two restricted stacks
Abstract Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the…
Étude de statistiques combinatoires et de leur impact en optimisation évolutionnaire
This thesis studies combinatorial objects, with both an algorithmic and a combinatorial point of view. In the combinatorial part, we take care first, the enumeration of Catalan words avoiding pairs of patterns of length three, presenting the proofs of each case with various enumeration methods. Catalan words are particular growth-restricted words counted by the eponymous integer sequence. More precisely, we systematically explore the structural properties of the sets of words under consideration and give enumerating results by constructive bijections or bivariate generating functions with respect to the length and descent number. Then, we study a sorting machine using two stacks in s…
Catalan words avoiding pairs of length three patterns
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 kn…
Transmission of Genetic Properties in Permutation Problems: Study of Lehmer Code and Inversion Table Encoding
Solution encoding describes the way decision variables are represented. In the case of permutation problems, the classical encoding should ensure that there are no duplicates. During crossover operations, repairs may be carried out to correct or avoid repetitions. The use of indirect encoding aims to define bijections between the classical permutation and a different representation of the decision variables. These encodings are not sensitive to duplicates. However, they lead to a loss of genetic properties during crossbreeding. This paper proposes a study of the impact of this loss both in the space of decision variables and in that of fitness values. We consider two indirect encoding: the …