Combinatoire énumérative et algébrique autour du PASEP

Abstract : This thesis comes within the scope of enumerative and algebraic combinatorics and studies the probabilities of the partially asymmetric exclusion process (PASEP).First, we bijectively prove a conjecture of Novelli-Thibon-Williams concerning the combinatorial interpretation of the entries of the transition matrices between some bases of the noncommutative symmetric functions algebra. More precisely, these matrices correspond to the transition matrices of, on the one hand the complete and ribbon bases and on the other hand the monomial and fundamental bases, both introduced by Tevlin. The coefficients of these matrices provide a refinement of the probabilities of the PASEP and are described using new statistics on permutations. This conjecture states that this refinement can also be described using classical statistics of the PASEP. In the second part, we study a generalization of the PASEP using two kinds of particles: the 2-PASEP. Hence, we give several combinatorial interpretations of the probabilities of this model. In order to do so, we define a new family of paths generalizing the Laguerre histories: the marked Laguerre histories. We also generalize the Françon-Viennot bijection between Laguerre histories and permutations to define partially signed permutations giving another combinatorial interpretation of these probabilities. In a third part, we generalize Tevlin's work in order to define a monomial basis and a fundamental basis on the algebra over segmented compositions. In order to describe the transition matrices between these bases and other bases already known in this algebra, we define an algebra indexed by partially signed permutations using the statistics previously defined to describe the combinatorics of the 2-PASEP. We also define some q-analogues of these bases related to the probabilities of the 2-PASEP according to the q parameter of this model. Finally, using the fact that partially signed permutations and segmented permutations are in bijection, we use the statistics defined previously to define descents on these objects and get a generalization of the Eulerian polynomials on segmented permutations. To study these polynomials, we use the algebraic tools introduced in the previous part
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Arthur Nunge. Combinatoire énumérative et algébrique autour du PASEP. Combinatoire [math.CO]. Université Paris-Est, 2018. Français. ⟨NNT : 2018PESC1116⟩. ⟨tel-02085841⟩

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