  A new Z-basis for the space of quasisymmetric functions (QSym, for short) is
presented. It is shown to have nonnegative structure constants, and several
interesting properties relative to the space of quasisymmetric functions
associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and
Reiner. In particular, for loopless matroids, this basis reflects the grading
by matroid rank, as well as by the size of the ground set. It is shown that the
morphism F is injective on the set of rank two matroids, and that
decomposability of the quasisymmetric function of a rank two matroid mirrors
the decomposability of its base polytope. An affirmative answer is given to the
Hilbert basis question raised by Billera, Jia, and Reiner.
