The twisted descent algebra and related Hopf algebras

I am going to report on joint work with Frederic Patras on the (free) twisted descent algebra. It has basis indexed by set compositions. Such set compositions are in 1-1 correspondence with monomials in non-commuting variables, with increasing trees, with faces of permutahedra, etc. These combinatorial correspondences extend to the level of Hopf algebras and provide links to the work of various authors during the past years, including Bergeron/Zabrocki, Chapoton and Novelli/Thibon.