A Faà di Bruno subalgebra of the graded dual of word symmetric functions

We show that the graded dual of the Hopf algebra of word symmetric functions contains the Faà di Bruno algebra as a Hopf subalgebra. In this context, we give the expansion of natural generators of the Faà di Bruno algebra on a natural basis of this graded dual. We also give a combinatorial interpretation for the coefficients arising in this expansion.