Integral majorization polytopes

The majorization polytope M(a) consists of all vectors dominated (or majorized, to be precise) by a given vector a ∈ ℝn; this is a polytope with extreme points being the permutations of a. For integral vector a, let ν(a) be the number of integral vectors contained in M(a). We present several properties of the function ν and provide an algorithm for computing ν(a)