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Constructions par greffe, combinatoire analytique et génération analytique

Abstract : Analytic combinatorics is a field which consist in applying methods from complex ana- lysis to combinatorial classes in order to obtain results on their asymptotic properties. We use for that specifications, which are a way to formalise the (often recursive) structure of the objects. In this thesis, we mainly devote ourselves to find new specifications for some combinatorial classes, in order to then apply more effective enumerative or random sampling methods. Indeed, for one combinatorial class several different specifications, based on different decompositions, may exist, making the classical methods - of asymptotic enu- meration or random sampling - more or less adapted. The first set of presented results focuses on Rémy’s algorithm and its underlying holonomic specification, based on a grafting operator. We develop a new and more efficient random sampler of binary trees and a random sampler of Motzkin trees based on the same principle. We then address some question relative to the study of subclasses of λ-terms. Finally, we present two other sets of results, on automatic specification of trees where occurrences of a given pattern are marked and on the asymptotic behaviour and the random sampling of digitally convex polyominoes. In every case, the new specifications give access to methods which could not be applied previously and lead to numerous new results.
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Submitted on : Tuesday, December 15, 2015 - 5:32:13 PM
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  • HAL Id : tel-01244436, version 1


Alice Jacquot. Constructions par greffe, combinatoire analytique et génération analytique. Analyse numérique [cs.NA]. Université Paris-Nord - Paris XIII, 2014. Français. ⟨NNT : 2014PA132014⟩. ⟨tel-01244436⟩



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