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Algèbres de Hopf combinatoires

Abstract : This thesis is in the field of algebraic combinatorics. In other words, the idea is to use algebraic structures, in this case of combinatorial Hopf algebras, to better study and understand the combinatorial objects and algorithms for composition and decomposition about these objects. This research is based on the construction and study of algebraic structure of combinatorial objects generalizing permutations. After recalling the background and notations of various objects involved in this research, we propose, in the second part, the study of the Hopf algebra introduced by Aguiar and Orellana based on uniform block permutations. By focusing on a description of these objects via well-known objects, permutations and set partitions, we propose a polynomial realization and an easier study of this algebra. The third section considers a second generalization interpreting permutations as matrices. We define and then study the families of square matrices on which we define algorithms for composition and decomposition. The fourth part deals with alternating sign matrices. Having defined the Hopf algebra of these matrices, we study the statistics and the behavior of the algebraic structure with these statistics. All these chapters rely heavily on computer exploration, and is the subject of an implementation using Sage software. This last chapter is dedicated to the discovery and manipulation of algebraic structures on Sage. We conclude by explaining the improvements to the study of algebraic structure through the Sage software
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Submitted on : Monday, June 27, 2016 - 4:48:18 PM
Last modification on : Saturday, January 15, 2022 - 3:56:08 AM


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  • HAL Id : tel-01338011, version 1


Rémi Maurice. Algèbres de Hopf combinatoires. Informatique et langage [cs.CL]. Université Paris-Est, 2013. Français. ⟨NNT : 2013PEST1196⟩. ⟨tel-01338011⟩



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