Ehrhart series

The Ehrhart series is a mathematical tool used to count the number of lattice points (points with integer coordinates) inside scaled versions of a geometric shape called a polytope. When you multiply a polytope by a positive integer (scaling it), the Ehrhart series encodes how many lattice points are contained within each scaled shape. It is represented as a generating function—a kind of formula—that compactly captures this counting sequence, revealing patterns and properties of the shape's structure related to the distribution of lattice points as it grows.