Symmetry-itemized enumeration of quadruplets of RS-stereoisomers. II. The partial-cycle-index method of the USCI approach combined with the stereoisogram approach

Abstract

The symmetry-itemized enumeration of quadruplets of stereoisograms is discussed by starting from a tetrahedral skeleton, where the partial-cycle-index (PCI) method of the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration of chemistry. Springer, Berlin, 1991) is combined with the stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004, Tetrahedron 60:11629–11638, 2004). Such a tetrahedral skeleton as contained in the quadruplet of a stereoisogram belongs to an RS-stereoisomeric group denoted by $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ , where the four positions of the tetrahedral skeleton accommodate achiral and chiral proligands to give quadruplets belonging to subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ according to the stereoisogram approach. The numbers of quadruplets are calculated as generating functions by applying the PCI method. They are itemized in terms of subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ , which are further categorized into five types. Quadruples for stereoisograms of types I–V are ascribed to subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ , where their features are discussed in comparison between RS-stereoisomeric groups and point groups.