Nonsymmetric operads in combinatorics

TL;DR

This paper explores the theory of nonsymmetric operads from a combinatorial perspective, highlighting their applications in computer science and combinatorics, and presenting modern algebraic tools and examples.

Contribution

It provides a comprehensive, modern overview of nonsymmetric operads with a focus on combinatorial applications and connections to algebraic structures.

Findings

Numerous combinatorial examples of operads are studied.

Connections between operads and algebraic structures are established.

The modern formal power series approach to operads is developed.

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.