The Unit-Subduced-Cycle-Index Methods and the Characteristic-Monomial Method. Their Relationship as Group-Theoretical Tools for Chemical Combinatorics

Among the four methods of the unit-subduced-cycle-index (USCI) approach, the subduced-cycle-index (SCI) method and the partial-cycle-index (PCI) method have been discussed by using adamantane of T_d-symmetry as a probe for enumeration problems, where USCIs are derived on the basis of permutaion representations, coset representations (CRs) and marks. After the examination of the SCIs and PCIs, Pólya's theorem that is a standard method of chemical combinatorics has been derived from the USCI approach. As another approach, a new method called the characteristic-monomial (CM) method has been developed by virtue of charactereistic monomials (CMs). The CMs have been derived from Q-conjugacy representations and Q-conjugacy characters, which have been related to irreducible representations and irreducible characters of the standard repertoire of chemical group theory. The two approaches have been compared to discuss group-theoretical tools for chemical combinatorics on a common basis.