# CM - Operads in combinatorics - M2 informatique fondamentale

## Abstract

Informally, an operad is a space of operations having one output and
several inputs that can be composed. Each operad leads to the
definition of category of algebras. This theory offers a tool to
study situations wherein several operations interact with each others.

This lecture begins by presenting some elementary objects of algebraic
combinatorics: combinatorial classes and combinatorial algebras. We
introduce then (non-symmetric) operads and study some tools allowing
to establish presentations by generators and relations of operads.
Koszul duality in non-symmetric operads is an important part of this
theory which shall be presented.

We end this lecture by reviewing some generalizations: colored operads,
symmetric operads, and pros. We shall also explain how the theory of
operads offers a tool to obtain enumerative results.

## Lectures

Here are the main presented notions:

**Lecture 1, January 16, 2019:**
combinatorial collections; operations on combinatorial
collections.

**Lecture 2, January 23, 2019:**
treelike structures; syntax trees; operations on syntax trees.