  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.
