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Combinatoire du polynôme de Tutte et des cartes planaires

This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight $z$ per face and a weight $u$ per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps $C$ weighted by $T_C(u+1,1)$, where $T_C$ is the Tutte polynomial of $C$. We begin by a purely combinatorial characterization of the corresponding generating function, denoted by $F(z,u)$. We deduce from this that $F(z,u)$ is differentially algebraic in $z$, that is, satisfies a polynomial differential equation in $z$. Finally, for $u \geq -1$, we study the asymptotic behaviour of the $n$th coefficient of $F (z,u)$. We observe a phase transition at $0$, with a very unusual regime in $n^{-3}\ln^{-2} (n)$ for $u \in [-1,0[$, which testifies a new universality class for planar maps. In the second part, we propose a framework unifying the notions of activity used in the literature to describe the Tutte polynomial. The new notion of activity thereby defined is called $Δ$-activity. It gathers all the notions of activities that were already known and has nice properties, as Crapo's property that defines a partition of the lattice of the spanning subgraphs into intervals with respect to the activity. Lastly we conjecture that every activity that describes the Tutte polynomial and that satisfies Crapo's property can be defined in terms of $Δ$-activity.