Type-itemized enumeration of quadruplets of RS-stereoisomers: II—cycle indices with chirality fittingness modulated by type-V quadruplets

Abstract

Fujita’s proligand method developed originally for combinatorial enumeration under point groups (Fujita in Theor Chem Acc 113:73–79, 2005) has been extended to meet combinatorial enumeration under Fujita’s stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004). A new succinct method for type-itemized enumeration of quadruplets of RS-stereoisomers has been developed on the basis of cycle indices with chirality fittingness (CI–CF) modulated by type-V quadruplets, where CI–CFs for the subgroups of the corresponding RS-stereoisomeric group are used to calculate CI–CFs of the respective types (type I to type V). By introducing ligand-inventory functions to these CI–CFs, generating functions have been obtained to give the numbers of inequivalent quadruplets, which are itemized with respect to five types. The method using a modulated CI–CF is applied to enumerations based on an oxirane skeleton, an allene skeleton, and a tetrahedral skeleton. The relationship of RS-stereoisomers versus stereoisomers is discussed by referring to the enumeration results of oxirane derivatives.