  In the 1995 paper entitled "Noncommutative symmetric functions," Gelfand, et.
al. defined two noncommutative symmetric function analogues for the power sum
basis of the symmetric functions, along with analogues for the elementary and
the homogeneous bases. They did not consider the noncommutative symmetric power
sum duals in the quasisymmetric functions, which have since been explored only
in passing by Derksen and Malvenuto-Reutenauer. These two distinct
quasisymmetric power sum bases are the topic of this paper. In contrast to the
simplicity of the symmetric power sums, or the other well known bases of the
quasisymmetric functions, the quasisymmetric power sums have a more complex
combinatorial description. As a result, although symmetric function proofs
often translate directly to quasisymmetric analogues, this is not the case for
quasisymmetric power sums. Neither is there a model for working with the
quasisymmetric power sums in the work of Gelfand, et. al., which relies heavily
on quasi-determinants (which can only be exploited by duality for our purposes)
and is not particularly combinatorial in nature. This paper therefore offers a
first glimpse at working with these two relatively unstudied quasisymmetric
bases, avoiding duality where possible to encourage a previously unexplored
combinatorial understanding.
