Combinatoire algébrique des permutations et de leurs généralisations

Abstract : This thesis is at the crossroads between combinatorics and algebra. It studies some algebraic problems from a combinatorial point of view, and conversely, some combinatorial problems have an algebraic approach which enables us tosolve them. In the first part, some classical statistics on permutations are studied: the peaks, the valleys, the double rises, and the double descents. We show that we can build sub algebras and quotients of FQSym, an algebra which basis is indexed by permutations. Then, we study classical combinatorial sequences such as Gandhi polynomials, refinements of Genocchi numbers, and Euler numbers in a non commutative way. In particular, we see that combinatorial interpretations arise naturally from the non commutative approach. Finally, we solve some freeness problems about dendriform algebras, tridendriform algebras and quadrialgebras thanks to combinatorics of some labelled trees
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Vincent Vong. Combinatoire algébrique des permutations et de leurs généralisations. Mathématiques générales [math.GM]. Université Paris-Est, 2014. Français. ⟨NNT : 2014PEST1185⟩. ⟨tel-01329402⟩

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