Coloring QSym and invariants of Fuss-Catalan Algebras

Descent compositions yield to the remarkable and well-studied Hopf subalgebras NSym of the Malvenuto-Reutenauer Hopf algebra SSym. These algebras can be obtained by a nice combinatorial construction: the standardized permutation of a word yields to a realization into words of SSym. Letting the variables be commutative gives a morphism from SSym to QSym. This is the core of the theory of noncommutative symmetric functions. When restricted to finitely many variables, QSym[x1,...,xn], can be understood as polynomial invariants/coinvariants of the Temperley-Lieb algebras. This was the work of Hivert on one part and Aval-Bergeron(s) on the other.