 This connection between the number of linear extensions and volume can be used to approximate the number of linear extensions of any partial order efficiently (despite the fact that computing this number exactly is #P-complete) by applying a randomized polynomial-time approximation scheme for polytope volume.The Ehrhart polynomial of the order polytope is a polynomial whose values at integer values formula_31 give the number of integer points in a copy of the polytope scaled by a factor of formula_31.For the order polytope, the Ehrhart polynomial equals (after a minor change of variables) the order polynomial of the corresponding partially ordered set.