Combinatorial enumeration of three-dimensional trees as stereochemical models of alkanes: an approach based on Fujita’s proligand method and dual recognition as uninuclear and binuclear promolecules

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Three-dimensional (3D) trees, which are defined as a 3D extension of trees, are enumerated by Fujita’s proligand method (Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86; 115 (2006) 37–53). Such 3D-trees are dually recognized as uninuclear promolecules and as binuclear ones. The 3D-trees regarded as uninuclear promolecules are enumerated to give the gross number of 3D-trees, which suffers from redundancy due to contaminants. To evaluate the number of such contaminants, the 3D-trees are alternatively enumerated as binuclear promolecules. Cycle indices with chirality fittingness (CI-CFs) composed of three kinds of sphericity indices (SIs), i.e., a d for homospheric cycles, c d for enantiospheric cycles, and b d for hemispheric cycles are obtained for evaluating promolecules of the two kinds. The CI-CFs for the uninuclear promolecules and those for the related binuclear promolecules are compared in terms of the dichotomy between balanced 3D-trees and unbalanced 3D-trees. Thereby, the redundancy due to such contaminants is deleted effectively so as to give the net number of 3D-trees. The validity of this procedure is proved in three ways, all of which are based on the respective modes of the correspondence between uninuclear promolecules and binuclear ones. In order to enumerate 3D-trees by following this procedure, the CI-CFs are converted into functional equations by substituting the SIs for a(x d ), c(x d ), and b(x d ). Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated under various conditions and collected up to 20 carbon content in a tabular form. Now, the stereochemical problems (on the number of stereoisomers) by van’t Hoff (van’t~Hoff, Arch. Neerlandaises des Sci. Exactes et Nat, 9 (1874) 445–454”) and by LeBel (LeBel, Bull. Soc. Chim. Fr. (2), 22 (1874) 337–347) and the enumeration problems (on the number of trees) by Cayley (Cayley, Philos. Mag. 47 (1874) 444–446), both initiated in the 1870s, have been solved in a common theoretical framework, which satisfies both chemical and mathematical requirements.