Nonsymmetric operads in combinatorics
arXiv:2104.12398 · doi:10.1007/978-3-030-02074-3
Abstract
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this theory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combinatorics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads 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. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmetric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics.
arXiv comment152 pages
References in corpus (26)
- Homology of generalized partition posets
- Two-dimensional topological gravity and equivariant cohomology
- Pre-Lie algebras and the rooted trees operad
- Algebraic and combinatorial structures on pairs of twin binary trees
- New identities in dendriform algebras
- Cambrian Hopf Algebras
- The Hopf algebra of diagonal rectangulations
- Sur le nombre d'intervalles dans les treillis de Tamari
- On some anticyclic operads
- Relating two Hopf algebras built from an operad
- On generating series of finitely presented operads
- Trialgebras and families of polytopes
- Order structure on the algebra of permutations and of planar binary trees
- The associative operad and the weak order on the symmetric groups
- Operads from posets and Koszul duality
- Completing the operadic butterfly
- Pluriassociative algebras I: The pluriassociative operad
- Pluriassociative algebras II: The polydendriform operad and related operads
- Colored operads, series on colored operads, and combinatorial generating systems
- The non-symmetric operad pre-Lie is free
- Duplicial algebras and Lagrange inversion
- Tree series and pattern avoidance in syntax trees
- Combalgebraic structures on decorated cliques
- Some Combinatorial Operators in Language Theory
- Examples of Com-PreLie Hopf algebras
- Confluence laws and Hopf-Borel type theorem for operads
Cited by in corpus (9)
- Celebrating Loday's Associahedron
- Tree series and pattern avoidance in syntax trees
- Generation of musical patterns through operads
- Graph insertion operads
- Cliff operads: a hierarchy of operads on words
- The music box operad: Random generation of musical phrases from patterns
- Decimation and Interleaving Operations in One-Sided Symbolic Dynamics
- Duality of graded graphs through operads
- Clones of pigmented words and realizations of special classes of monoids