Mathematics – Combinatorics
Scientific paper
2007-04-06
Journal of Combinatorial Theory, Series A 115 (2008) 777-798
Mathematics
Combinatorics
25 pages; exposition tightened, typos corrected; to appear in the Journal of Combinatorial Theory, Series A
Scientific paper
10.1016/j.jcta.2007.10.003
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.
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