 More geometrically, we may list the elements formula_57, and identify any mapping formula_58 with the point formula_59; then the order polytope is the set of points formula_60 with formula_61 if formula_62.The Ehrhart polynomial counts the number of integer lattice points inside the dilations of a polytope.Specifically, consider the lattice formula_63 and a formula_64-dimensional polytope formula_65 with vertices in formula_66; then we define

the number of lattice points in formula_68, the dilation of formula_69 by a positive integer scalar formula_1.