  We develop the theory of weighted P-partitions, which generalises the theory
of P-partitions from labelled posets to weighted labelled posets. We define the
related generating functions in the natural way and compute their product,
coproduct and other properties. As an application we introduce the basis of
combinatorial power sums for the Hopf algebra of quasisymmetric functions and
the reverse basis, both of which refine the power sum symmetric functions.
These bases share many properties with the type 1 and type 2 quasisymmetric
power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and
moreover expand into the monomial basis of quasisymmetric functions with
nonnegative integer coefficients. We prove formulas for products, coproducts
and classical quasisymmetric involutions via the combinatorics of P-partitions,
and give combinatorial interpretations for the coefficients when expanded into
the monomial and fundamental bases.
