Configurations and parallelograms associated to centers of mass

Fix integers k and t . The t–fold center of mass arrangement M(t, k) for integers t with k ≥ t ≥ 1 is defined as the subspace of the k–fold product Ck given by ordered k–tuples of points (x1, . . . , xk) such that the centroids of any set of t elements in the underlying set {x1, . . . , xk} σt(xi1 , xi2 , . . . , xit ) = (1/t)(xi1 + xi2 + · · ·+ xit ) are distinct for all distinct subsets {xi1 , xi2 , . . . , xit}, and {xj1 , xj2 , . . . , xjt}. In particular, M(t, k) is the complement of the union of the hyperplanes specified by