Contents -- 1. Introduction -- 1.1 Two-Dimensional versus Three-Dimensional Structures -- 1.1.1 Two-Dimensional Structures in Early History of Organic Chemistry -- 1.1.2 Three-Dimensional Structures After Beginning of Stereochemistry -- 1.1.3 Arbitrary Switching Between 2D-Based and 3D-Based Concepts -- 1.2 Problematic Methodology for Categorizing Isomers and Stereoisomers -- 1.2.1 Same or Different -- 1.2.2 Dual Definition of Isomers -- 1.2.3 Positional Isomers as a Kind of Constitutional Isomers -- 1.3 Problematic Methodology for Categorizing Enantiomers and Diastereomers -- 1.3.1 Enantiomers -- 1.3.2 Diastereomers -- 1.3.3 Chirality and Stereogenicity -- 1.4 Total Misleading Features of the Traditional Terminology on Isomers -- 1.4.1 Total Misleading Flowcharts -- 1.4.2 Another Flowchart With Partial Solutions -- 1.4.3 More Promising Way -- 1.5 Isomer Numbers -- 1.5.1 Combinatorial Enumeration as 2D Structures -- 1.5.2 Importance of the Proligand-Promolecule Model -- 1.5.3 Combinatorial Enumeration as 3D Structures -- 1.6 Stereoisograms -- 1.6.1 Stereoisograms as Diagrammatic Expressions of RS-Stereoisomeric Groups -- 1.6.2 Theoretical Foundations and Group Hierarchy -- 1.6.3 Avoidance of Misleading Standpoints of R/S-Stereodescriptors -- 1.6.4 Avoidance of Misleading Standpoints of pro-R/pro-S-Descriptors -- 1.6.5 Global Symmetries and Local Symmetries -- 1.6.6 Enumeration under RS-Stereoisomeric Groups -- 1.7 Aims of Mathematical Stereochemistry -- References -- 2. Classification of Isomers -- 2.1 Equivalence Relationships of Various Levels of Isomerism -- 2.1.1 Equivalence Relationships and Equivalence Classes -- 2.1.2 Enantiomers, Stereoisomers, and Isomers -- 2.1.3 Inequivalence Relationships -- 2.1.4 Isoskeletomers as a Missing Link for Consistent Terminology.

2.1.5 Constitutionally-Anisomeric Relationships vs. Constitutionally-Isomeric Relationships -- 2.2 Revised Flowchart for Categorizing Isomers -- 2.2.1 Design of a Revised Flowchart for Categorizing Isomers -- 2.2.2 Illustrative Examples -- 2.2.3 Restriction of the Domain of Isomerism -- 2.2.4 Harmonization of 3D-Based Concepts with 2D-Based Concepts -- References -- 3. Point-Group Symmetry -- 3.1 Stereoskeletons and the Proligand-Promolecule Model -- 3.1.1 Configuration and Conformation -- 3.1.2 The Proligand-Promolecule Model -- 3.2 Point Groups -- 3.2.1 Symmetry Axes and Symmetry Operations -- 3.2.2 Construction of Point Groups -- 3.2.3 Subgroups of a Point Group -- 3.2.4 Maximum Chiral Subgroup of a Point Group -- 3.2.5 Global and Local Point-Group Symmetries -- 3.3 Point-Group Symmetries of Stereoskeletons -- 3.3.1 Stereoskeletons of Ligancy 4 -- 3.3.2 Stereoskeletons of Ligancy 6 -- 3.3.3 Stereoskeletons of Ligancy 8 -- 3.3.4 Stereoskeletons Having Two or More Orbits -- 3.4 Point-Group Symmetries of (Pro)molecules -- 3.4.1 Derivation of Molecules from a Stereoskeleton via Promolecules -- 3.4.2 Orbits in Molecules and Promolecules Derived from Stereoskeletons -- 3.4.3 The SCR Notation -- 3.4.4 Site Symmetries vs. Coset Representations for Symmetry Notations -- References -- 4. Sphericities of Orbits and Prochirality -- 4.1 Sphericities of Orbits -- 4.1.1 Orbits of Equivalent Proligands -- 4.1.2 Three Kinds of Sphericities -- 4.1.3 Chirality Fittingness for Three Modes of Accommodation -- 4.2 Prochirality -- 4.2.1 Confusion on the Term 'Prochirality' -- 4.2.2 Prochirality as a Geometric Concept -- 4.2.3 Enantiospheric Orbits vs. Enantiotopic Relationships -- 4.2.4 Chirogenic Sites in an Enantiospheric Orbit -- 4.2.5 Prochirality Concerning Chiral Proligands in Isolation.

4.2.6 Global Prochirality and Local Prochirality -- References -- 5. Foundations of Enumeration Under Point Groups -- 5.1 Orbits Governed by Coset Representations -- 5.1.1 Coset Representations -- 5.1.2 Mark Tables -- 5.1.3 Multiplicities of Orbits -- 5.2 Subduction of Coset Representations -- 5.2.1 Subduced Representations -- 5.2.2 Unit Subduced Cylce Indices (USCIs) -- References -- 6. Symmetry-Itemized Enumeration Under Point Groups -- 6.1 Fujita's USCI Approach -- 6.1.1 Historical Comments -- 6.1.2 USCI-CFs for Itemized Enumeration -- 6.1.3 Subduced Cycle Indices for Itemized Enumeration -- 6.2 The FPM Method of Fujita's USCI Approach -- 6.2.1 Fixed-Point Vectors (FPVs) and Multiplicity Vectors (MVs) -- 6.2.2 Fixed-Point Matrices (FPMs) and Isomer-Counting Matrices (ICMs) -- 6.2.3 Practices of the FPM Method -- 6.3 The PCI Method of Fujita's USCI Approach -- 6.3.1 Partial Cycle Indices With Chirality Fittingness (PCI-CFs) -- 6.3.2 Partial Cycle Indices Without Chirality Fittingness (PCIs) -- 6.3.3 Practices of the PCI Method -- 6.4 Other Methods of Fujita's USCI Approach -- 6.4.1 The Elementary-Superposition Method -- 6.4.2 The Partial-Superposition Method -- 6.5 Applications of Fujita's USCI Approach -- 6.5.1 Enumeration of Flexible Molecules -- 6.5.2 Enumeration of Molecules Interesting Stereochemically -- 6.5.3 Enumeration of Inorganic Complexes -- 6.5.4 Enumeration of Organic Reactions -- References -- 7. Gross Enumeration Under Point Groups -- 7.1 Counting Orbits -- 7.2 Pólya's Theorem of Counting -- 7.3 Fujita's Proligand Method of Counting -- 7.3.1 Historical Comments -- 7.3.2 Sphericities of Cycles -- 7.3.3 Products of Sphericity Indices -- 7.3.4 Practices of Fujita's Proligand Method -- 7.3.5 Enumeration of Achiral and Chiral Promolecules -- References.

8. Enumeration of Alkanes as 3D Structures -- 8.1 Surveys With Historical Comments -- 8.2 Enumeration of Alkyl Ligands as 3D Planted Trees -- 8.2.1 Enumeration of Methyl Proligands as Planted Promolecules -- 8.2.2 Recursive Enumeration of Alkyl ligands as Planted Promolecules -- 8.2.3 Functional Equations for Recursive Enumeration of Alkyl ligands -- 8.2.4 Achiral Alkyl Ligands and Pairs of Enantiomeric Alkyl Ligands -- 8.3 Enumeration of Alkyl Ligands as Planted Trees -- 8.3.1 Alkyl Ligands or Monosubstituted Alkanes as Graphs -- 8.3.2 3D Structures vs. Graphs for Characterizing Alkyl Ligands or Monosubstituted Alkanes -- 8.4 Enumeration of Alkanes (3D-Trees) as 3D-Structural Isomers -- 8.4.1 Alkanes as Centroidal and Bicentroidal 3D-Trees -- 8.4.2 Enumeration of Centroidal Alkanes (3D-Trees) as 3D-Structural Isomers -- 8.4.3 Enumeration of Bicentroidal Alkanes (3D-Trees) as 3D-Structural Isomers -- 8.4.4 Total Enumeration of Alkanes as 3D-Trees -- 8.5 Enumeration of Alkanes (3D-Trees) as Steric Isomers -- 8.5.1 Centroidal Alkanes (3D-Trees) as Steric Isomers -- 8.5.2 Bicentroidal Alkanes (3D-Trees) as Steric Isomers -- 8.5.3 Total Enumeration of Alkanes (3D-Trees) as Steric Isomers -- 8.6 Enumeration of Alkanes (Trees) as Graphs or Constitutional Isomers -- 8.6.1 Alkanes as Centroidal and Bicentroidal Trees -- 8.6.2 Enumeration of Centroidal Alkanes (Trees) as Constitutional Isomers -- 8.6.3 Enumeration of Bicentroidal Alkanes (Trees) as Constitutional Isomers -- 8.6.4 Total Enumeration of Alkanes (Trees) as Graphs or Constitutional Isomers -- References -- 9. Permutation-Group Symmetry -- 9.1 Historical Comments -- 9.2 Permutation Groups -- 9.2.1 Permutation Groups as Subgroups of Symmetric Groups -- 9.2.2 Permutations vs. Reflections -- 9.3 RS-Permutation Groups.

9.3.1 RS-Permutations and RS-Diastereomeric Relationships -- 9.3.2 RS-Permutation Groups vs. Point Groups -- 9.3.3 Formulation of RS-Permutation Groups -- 9.3.4 Action of RS-Permutation Groups -- 9.3.5 Misleading Features of the Conventional Terminology -- 9.4 RS-Permutation Groups for Skeletons of Ligancy 4 -- 9.4.1 RS-Permutation Group for a Tetrahedral Skeleton -- 9.4.2 RS-Permutation Group for an Allene Skeleton -- 9.4.3 RS-Permutation Group for an Ethylene Skeleton -- References -- 10. Stereoisograms and RS-Stereoisomers -- 10.1 Stereoisograms as Integrated Diagrammatic Expressions -- 10.1.1 Elementary Stereoisograms of Skeletons with Position Numbering -- 10.1.2 Stereoisograms Based on Elementary Stereoisograms -- 10.2 Enumeration Under RS-Stereoisomeric Groups -- 10.2.1 Subgroups of the RS-Stereoisomeric Group C3v sI -- 10.2.2 Coset Representations -- 10.2.3 Mark Table and its Inverse -- 10.2.4 Subduction for RS-Stereoisomeric Groups -- 10.2.5 USCI-CFs for RS-Stereoisomeric Groups -- 10.2.6 SCI-CFs for RS-Stereoisomeric Groups -- 10.2.7 The PCI Method for RS-Stereoisomeric Groups -- 10.2.8 Type-Itemized Enumeration by the PCI Method -- 10.2.9 Gross Enumeration Under RS-Stereoisomeric Groups -- 10.3 Comparison with Enumeration Under Subgroups -- 10.3.1 Comparison with Enumeration Under Point Groups -- 10.3.2 Comparison with Enumeration Under RS-Permutation Groups -- 10.3.3 Comparison with Enumeration Under Maximum-Chiral Point Subgroups -- 10.4 RS-Stereoisomers as Intermediate Concepts -- References -- 11. Stereoisograms for Tetrahedral Derivatives -- 11.1 RS-Stereoisomeric Group Td sI and Elementary Stereoisogram -- 11.2 Stereoisograms of Five Types for Tetrahedral Derivatives -- 11.2.1 Type-I Stereoisograms of Tetrahedral Derivatives.

11.2.2 Type-II Stereoisograms of Tetrahedral Derivatives.