Cyclic Symmetry of the Scaled Simplex

Let Z_m^k consist of the m^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A_k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order (k+1), so does Z_m^k. By bijectively exchanging the natural poset structure of Z_m^k for a natural cyclic action on a set of words, we prove an instance of the cyclic sieving phenomenon.