Given a simple graph G from the first stage, the second stage assigns elements to vertices in all distinct ways. The element counts must be correct, and we must have valence degree at each ≥ vertex. More onerously, we only want one member of each equivalence class of element assignment under the automorphism group of G. We next explain how this is accomplished. The vertices of G are arbitrarily numbered 1,2,...,n. An element assignment can be represented as a list showing the element assigned to each vertex in order of vertex number. For example, a valid list might be L = (C,C,C,S,O,C,C,C,O,C). Automorphisms of G have an action on lists that permutes their entries. Namely, for list L and automorphism the list (L) assigns the same element to vertex (v) as L assigns to v, for each γ, γ γ vertex v. Thus,