  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.
