# Habilitation à diriger les recherches

### Title

**Operads in algebraic combinatorics**
(Opérades en combinatoire algébrique)

### Abstract

This habilitation thesis fits in the fields of algebraic and
enumerative combinatorics, with connections with computer science.
The main ideas developed in this work consist in endowing
combinatorial objects (words, permutations, trees, integer
partitions, 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 by generators and relations,
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 (functorial or not) 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 pre-Lie 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 text is organized as follows. The first two chapters pose the
elementary notions of combinatorics and algebraic combinatorics
used in the whole work. The last ten chapters contain our original
research results fitting the context presented above.

### Keys words and phrases

Algebraic combinatorics; Computer science; Formal power series;
Tree; Rewrite system; Poset; Operad; Colored operad; Hopf bialgebra;
Pro.

### Composition du jury

- Frédéric Chapoton (Président du jury)

- Jean-Yves Thibon (Garant d'habilitation)

- Pierre-Louis Curien (Rapporteur)
- Loïc Foissy (Rapporteur)
- Dominique Manchon (Rapporteur)

- Alessandra Frabetti (Examinateur)
- Florent Hivert (Examinateur)
- Jean-Gabriel Luque (Examinateur)

### Soutenance

Le diaporama de la soutenance se trouve
ici.

### Manuscrit d'habilitation

Il se trouve ici.

### Détails pratiques

La soutenance a eu lieu

le lundi 4 décembre 2017 à 14 h 00
en salle de séminaire (4B05R)
à l'université Paris-Est Marne-la-Vallée.
Voici un
plan d'accès.