  We investigate quasisymmetric functions coming from combinatorial Hopf
monoids. We show that these invariants arise naturally in Ehrhart theory, and
that some of their specializations are Hilbert functions for relative
simplicial complexes. This class of complexes, called forbidden composition
complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf
algebras, Ehrhart theory, and commutative algebra. We also study various
specializations of quasisymmetric functions.
