Sur l'\'enum\'eration de structures discr\`etes, une approche par la th\'eorie des relations

Abstract

Theory of relations is the framework of this thesis. It is about enumeration of finite structures. Let $\mathscr C$ be a class of finite combinatorial structures, the \emph{profile} of $\mathscr C$ is the function $\varphi_{\mathscr C}$ which count, for every $n$, the number of members of $\mathscr{C}$ defined on $n$ elements, isomorphic structures been identified. The generating function for $\mathscr C$ is $\mathcal H_{\mathscr C}(X):=\sum_{n\geqq 0}\varphi_{\mathscr C}(n)X^n$. Many results about the behavior of the function $\varphi_{\mathscr C}$ have been obtained. Albert and Atkinson have shown that the generating series of the profile of some classes of permutations are algebraic. we show how this result extends to classes of ordered binary structures using the notions of theory of relations. This is the subject of the first part of this thesis. The second part is concerned with the notion of minimality. An hereditary class of finite structures is minimal if it is infinite and every proper hereditary subclass is finite. We show, in particular, that ind-minimal classes are wqo ages and their number is the continuum. The last part is motivated by the surprising phenomenon of the \emph{jump} observed in the behavior of the profile of hereditary classes of finite structures. We show that the profile of an hereditary classe made of ordered structures which have finite monomorphic decomposition is a polynomial. We also show that if the profile of a hereditary class of finite ordered binary structures is not bounded by a polynomial then it is at least exponential. This result generalizes the result obtained by Balogh, Bollob\'as and Morris (2006) for ordered graphs.