Sphericities of Double Cosets, Double Coset Representations, and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a _d , c _d , or b _d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.