The enumeration and symmetry-significant properties of derivative lattices. III. Periodic colourings of a lattice

In the triclinic case, structures that can be described in terms of arrangements of a set number of possible subunits occupying the unit cells of an underlying lattice may be enumerated by their derivative lattice index n and stoichiometry, e.g. XmY(n−m) for two types of subunits. This process involves counting the number, H(n, m), of such patterns possible on the frame of the colour lattice group, followed by the elimination of any patterns that belong to a derivative lattice of lower index. The resulting numbers, K(n, m), then have the property K(n,m) \le (1/n)[_{m}^{n}] \le H(n,m)where \bigg[{m \over n}\bigg] is the binomial coefficient. These expressions are equalities if n and m are mutually prime. H(n, m) and K(n, m) are tabulated for all colour lattice groups n ≤ 20 and all corresponding values of m. The method may be extended to a larger number of subunit types by a simple change to the figure-generating function.