Enumeration of polytypes MX and MX₂ through the use of the symmetry of the Zhdanov symbol

The different close-packed polytypes MX and MX₂ have been enumerated for each of the possible space groups by counting the corresponding Zhdanov symbols for each space group and period of stacking, P, by the use of elementary combinatorial techniques. In special cases, simple closed formulae are obtained for these numbers as functions of P. The symmetry properties of the Zhdanov symbol have been investigated with the help of its cyclotomic representation and the two-color symmetry point group thereof. Zhdanov-like rules have been developed for MX₂ polytypes. The SiC cases have been generated to P = 18 under the `1-exclusion' rule and the possible diamond polytypes have been examined.