Operads in algebraic combinatorics

TL;DR

This thesis develops algebraic structures on combinatorial objects like words, permutations, and trees, enabling new insights and constructions in algebraic combinatorics, including operads, Hopf algebras, and enumerative methods.

Contribution

It introduces new algebraic constructions on combinatorial objects, such as functors from various algebraic structures to operads, and generalizes existing frameworks like the noncommutative Faà di Bruno Hopf algebra.

Findings

Constructed functors from monoids and posets to operads

Generalized the noncommutative Faà di Bruno Hopf algebra

Developed algebraic tools for combinatorial enumeration

Abstract

The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, 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, 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 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 Fa\`a 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 first two chapters pose the elementary notions of combinatorics and algebraic combinatorics used here. The last ten chapters contain our original research.