A matroid-friendly basis for the quasisymmetric functions

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 quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. 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 distinguishes isomorphism classes 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 to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.