Two problems on lattice point enumeration of rational polytopes

Motivated by the generalization of Ehrhart theory with group actions, the first part of this thesis makes progress towards obtaining the equivariant Ehrhart theory of the permutahedron. The subset that is fixed by a group action on the permutahedron is itself a rational polytope. We prove that these fixed polytopes are combinatorially equivalent to lower dimensional permutahedra. Furthermore, we show that these fixed polytopes are zonotopes, i.e., Minkowski sum of line segments. This part is joint work with Anna Schindler. The second part of this thesis provides a decomposition of the /i*-polynomial for rational polytopes. This decomposition is an analogue to the decomposition proven by Ulrich Betke and Peter McMullen for lattice polytopes.