Counting Lattice Points in Minkowski Sums of Cross Polytopes

Motivated by Postnikov's study of lattice-point enumeration in Minkowski sums of simplices, we investigate lattice points in Minkowski sums of cross polytopes and establish analogous results, together with several related consequences. In particular, we introduce the support-enumerator associated with Postnikov's notion of draconian sequences and show that it coincides with the $h^*$-polynomial of the corresponding root polytope. This provides a new interpretation of the $h^*$-polynomial and yields a simple method for computing the volume of the corresponding polytope. We further exploit the symmetry of such root polytopes to establish a duality property for support-enumerators, which in turn provides a proof of a conjecture by Chapoton and Athanasiadis concerning the $h$-polynomials of preorders. As an immediate consequence, we prove that a Minkowski sum of cross polytopes and its dual polytope have the same number of lattice points. This duality then leads to a general formula for the number of lattice points in Minkowski sums of cross polytopes in terms of draconian sequences. Our formula enables us to compute the Ehrhart polynomials of these polytopes and show that they are Ehrhart positive. Furthermore, the formula allows us to derive analogous formulas for the number of lattice points on their boundaries and for their surface volumes.