Found after some more searching:

LattE counts lattice points within rational polytopes via Barvinok's algorithm, which computes the rational multivariate generating function of the lattice points in a region (and then plugs in $x = 1$ to get the count). This works because the generating functions can be additively combined over different pieces of polytope (at vertices and faces). A paper describing the software is

De Loera et al., Effective lattice point counting in rational convex polytopes

Sage appears to use LattE as a dependency, so that's an easy way to access it.

Found after some more searching:

LattE counts lattice points within rational polytopes via Barvinok's algorithm, which computes the rational multivariate generating function of the lattice points in a region (and then plugs in $x = 1$ to get the count). This works because the generating functions can be additively combined over different pieces of the polytope (at vertices and faces). A paper describing the software is

De Loera et al., Effective lattice point counting in rational convex polytopes

Sage appears to use LattE as a dependency, so that's an easy way to access it.

Found after some more searching:

LattE counts lattice points within rational polytopes via Barvinok's algorithm, which computes the rational multivariate generating function of the lattice points in a region (and then plugs in $x = 1$ to get the count). This works because the generating functions can be additively combined over different pieces of polytope (at vertices and faces). A paper describing the software is

De Loera et al., Effective lattice point counting in rational convex polytopes

Found after some more searching:

LattE counts lattice points within rational polytopes via Barvinok's algorithm, which computes the rational multivariate generating function of the lattice points in a region (and then plugs in $x = 1$ to get the count). This works because the generating functions can be additively combined over different pieces of polytope (at vertices and faces). A paper describing the software is

De Loera et al., Effective lattice point counting in rational convex polytopes

Sage appears to use LattE as a dependency, so that's an easy way to access it.