Titlebook: Nonsymmetric Operads in Combinatorics; Samuele Giraudo Book 2018 Springer Nature Switzerland AG 2018 Combinatorics.Nonsymmetric operands.t

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s consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic,  and symmetric operads are reviewed.. .978-3-030-02074-3

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Algebraic Structures, presented. Most of the algebraic structures encountered in algebraic combinatorics like associative, dendriform, pre-Lie algebras, and Hopf bialgebras fit into this framework. This chapter contains classical examples of such structures.

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Nonsymmetric Operads,ous chapters. We consider here also set-operads, algebras over operads, free operads, presentations by generators and relations, Koszul duality and Koszulity of operads. At the end of the chapter, several examples of operads on a large family of combinatorial collections are provided.

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Introduction,Let us start this text by considering two fields of mathematics: combinatorics and algebra, and a field of computer science: programming. These topics have far more in common than it appears at first glance and we shall explain why.

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Enriched Collections,This preliminary chapter contains general notions about combinatorics used in the rest of the book. We introduce the notion of collections of combinatorial objects and then the notions of posets and rewrite systems, which are seen as collections endowed with some extra structure.

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Treelike Structures,This second chapter is devoted to present general notions about treelike structures. We present more precisely the ones appearing in the algebraic and combinatorial context of nonsymmetric operads. Rewrite systems of syntax trees are exposed, as well as methods to prove their termination and their confluence.

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Applications and Generalizations,This last chapter is devoted to review some applications of the theory of operads for enumerative prospects. To this aim, we present formal power series on operads, generalizing usual generating series. We also provide an overview on enrichments of operads: colored operads, cyclic operads, symmetric operads, and pros.