Invariants and Coinvariants of the Symmetric Group in Noncommuting   Variables
  We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions indexed by compositions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials. 
Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, Mike Zabrocki