Stereoisograms for three-membered heterocycles: IV. Half-size-subgroup method for type-itemized enumeration of oxiranes under an RS-stereoisomeric group

摘要

The half-size-subgroup method for type-itemized enumeration of stereoisograms has been applied to epoxide (oxirane) derivatives after introducing the basic terminology of Fujita's stereoisogram approach. Cycle indices with chirality fittingness (CI-CFs) for type-itemized enumerations of epoxide derivatives are calculated for the three half-size subgroups of an RS-stereoisomeric group (i.e., a point group, RS-stereogenic group, and LR-permutation group) along with a maximum normal subgroup. After ligand-inventory functions for enumerations are introduced into the CI-CFs, the resulting equations are expanded to give generating functions for type-itemized enumeration of stereoisograms. Three categories for characterizing five types of stereoisograms are recognized to obtain the numbers of derivatives: Category 1 for existence of type I and type IV; Category 2 for existence of type II, III, and V; and Category 3 for coexistence of Categories 1 and 2. A general procedure for solving category 3 is developed by separating into Categories 1 and 2. Thereafter, the half-size-subgroup method for type-itemized enumeration (not via symmetry-itemized enumeration) is discussed.