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We introduce a new algebraic framework based on the deformation of preL...
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Combinatorial Properties of Metrically Homogeneous Graphs
Ramsey theory looks for regularities in large objects. Model theory stud...
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Algebraic and algorithmic aspects of radical parametrizations
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Tuning as convex optimisation: a polynomial tuner for multiparametric combinatorial samplers
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Algorithmic and algebraic aspects of unshuffling permutations
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AlgebraicallyInformed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures
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Pell hyperbolas in DLPbased cryptosystems
We present a study on the use of Pell hyperbolas in cryptosystems with s...
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Operads in algebraic combinatorics
The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, representations), to collect combinatorial information about the underlying objects. The algebraic structures the most encountered here are magmas, posets, associative algebras, dendriform algebras, Hopf bialgebras, operads, and pros. This work explores the aforementioned research direction and provides many constructions having the particularity to build algebraic structures on combinatorial objects. We develop for instance a functor from nonsymmetric colored operads to nonsymmetric operads, from monoids to operads, from unitary magmas to nonsymmetric operads, from finite posets to nonsymmetric operads, from stiff pros to Hopf bialgebras, and from precompositions to nonsymmetric operads. These constructions bring alternative ways to describe already known structures and provide new ones, as for instance, some of the deformations of the noncommutative Faà di Bruno Hopf bialgebra of Foissy and a generalization of the dendriform operad of Loday. We also use algebraic structures to obtain enumerative results. In particular, nonsymmetric colored operads are promising devices to define formal series generalizing the usual ones. These series come with several products (for instance a preLie product, an associative product, and their Kleene stars) enriching the usual ones on classical power series. This provides a framework and a toolbox to strike combinatorial questions in an original way. The first two chapters pose the elementary notions of combinatorics and algebraic combinatorics used here. The last ten chapters contain our original research.
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