Computing parametric rational generating functions with a primal Barvinok algorithm

Computations with Barvinok’s short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion–exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space.