Unit Subduced Cycle Indices Research Articles

Application of GAP (groups, algorithms and programming) system to stereoisograms for characterizing RS-stereoisomers of cubane derivatives

The PCI (Partial-Cycle-Index) method of Fujita’s USCI (Unit-Subduced-CycleIndex) approach has been applied to symmetry-itemized enumerations of cubane derivatives, where groups for specifying three-aspects of symmetry, i.e., the point group for chirality/achirality, the RS-stereogenic group for RS-stereogenicity/RS-astereogenicity, and the LR-permutation group for sclerality/ascrelarity are considered as the subgroups of the RS-stereoisomeric group . Five types of stereoisograms are adopted as diagrammatical expressions of , after combined-permutation representations (CPR) are created as new tools for treating various groups according to Fujita’s stereoisogram approach. The use of CPRs under the GAP (Groups, Algorithms and Programming) system has provided new GAP functions for promoting symmetry-itemized enumerations. The type indices for characterizing stereoisograms (e.g., for a type-V stereoisogram) have been sophisticated into RS-stereoisomeric indices (e.g., for a cubane derivative with the composition ). The type-V stereoisograms for cubane derivatives with the composition are discussed under extended pseudoasymmetry as a new concept.

Standardization of Mark Tables and USCI-CF (Unit Subduced Cycle Indices with Chirality Fittingness) Tables Derived from Different D3h-Skeletons

Abstract Combined-permutation representations (CPRs) for characterizing D3h-skeletons (i.e., a cyclopropane skeleton, a trigonal bipyramidal skeleton, an iceane skeleton, and so on) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycles to give the CPR of degree 8 (= 6 + 2) for the cyclopropane skeleton, the CPR of degree 7 (= 5 + 2) for the trigonal bipyramidal skeleton, the CPR of degree 14 (= 12 + 2) for the iceane skeleton, and so on. Mark tables (tables of marks) of these CPRs are different in the alignment of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton by means of the newly-developed GAP functions, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

Mathematical Stereochemistry as the Theoretical Foundations of Organic and Inorganic Chemistry

Mathematical stereochemistry is discussed by surveying books written on Fujita's USCI (unit-subduced-cycle-index) approach, Fujita's proligand method, Fujita's stereoisogram approach, and related matters from a viewpoint of developing an interdisciplinary chemistry/mathematics field.

Half-Century Journey from Synthetic Organic Chemistry to Mathematical Stereochemistry through Chemoinformatics

My half-century journey started from synthetic organic chemistry. During the first stage of my journey, my interest in stereochemistry was initiated through the investigation on the participation of steric effects in reactive intermediates, cylophanes, strained heterocycles, and organic compounds for photography. In chemoinformatics as the next stage of the journey, I proposed the concept of imaginary transition structures (ITSs) as computer-oriented representation of organic reactions. My interest was stimulated to attack combinatorial enumeration through the investigation on enumeration of subgraphs of ITSs. Stereochemistry and combinatorial enumeration was combined in my interest, so that I reached mathematical stereochemistry as the final stage of my journey. Fujita's unit-subduced-cycle-index (USCI) approach, Fujita's proligand method, and Fujita's stereoisogram approach were developed, so as to integrate van't Hoff's way (asymmetry, stereogenicity) and Le Bel's way (dissymmetry, chirality), which caused continuous confusion in the history of stereochemistry.

Systematic Enumeration and Symmetries of Cubane Derivatives.

The feasibilities of Fujita's unit-subduced-cycle-index (USCI) approach, Fujita's proligand method, and Fujita's stereoisogram approach have been demonstrated by applying them to cubane derivatives as probes. They provide us with a new set of theoretical foundations for comprehensive investigation of geometric and stereoisomeric features of stereochemistry. The new set of theoretical foundations is based on mathematical formulations so as to explore mathematical stereochemistry as a new interdisciplinary field of stereochemistry.

Symmetry-Itemized Enumeration of Quadruplets of RS-Stereoisomers Derived from a Trigonal Pyramidal Skeleton

Abstract Stereoisograms of five types are devised as graphical expressions of the action of an RS-stereoisomeric group on a trigonal pyramidal skeleton. A quadruplet of promolecules (i.e., a reference promolecule, its enantiomer, its RS-diastereomer, and its holantimer) contained in a stereoisogram is counted once as an equivalence class under the action of an RS-stereoisomeric group. An RS-stereoisomeric group <img align="middle" src="./Graphics/abst-20150015-1.gif"/> is constructed by starting from the point group C3v of a trigonal pyramidal skeleton, where a point group C3v, an RS-permutation group <img align="middle" src="./Graphics/abst-20150015-2.gif"/>, a ligand-reflection group C3Î are integrated. Fujita’s unit-subduced-cycle-index (USCI) approach for enumeration under point groups is extended to cover enumeration under RS-stereoisomeric groups. The fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method of the USCI approach are applied to symmetry-itemized and type-itemized enumeration of quadruplets of promolecules under the action of the RS-stereoisomeric group <img align="middle" src="./Graphics/abst-20150015-1.gif"/>. Theoretical foundations of stereochemical nomenclature are discussed on the basis of Fujita’s stereoisogram approach, where three aspects of absolute configuration are emphasized. Prochirality and pro-RS-stereogenicity are clarified to be conceptually distinct, just as chirality and RS-stereogenicity are conceptually distinct. Young’s tableaux for (pro)chirality and those for (pro-)RS-stereogenicity are compared to demonstrate such conceptual distinctions. It follows that a pair of R/S-stereodescriptors is assigned on the basis of RS-stereogenicity, not of chirality; and a pair of pro-R/pro-S-descriptors is assigned on the basis of pro-RS-stereogenicity, not of prochirality.

Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\). The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\), which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\). According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\). After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

The RS-stereoisomeric group $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group $$\mathbf{O}_{h}$$ of order 48. The non-redundant set of subgroups (SSG) of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is obtained by referring to the non-redundant SSG of $$\mathbf{O}_{h}$$ . The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ , which is closely related to the $$\mathbf{O}_{h}(/\mathbf{D}_{3d})$$ . According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ . After the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Elementary-Superposition Method of the USCI Approach

Abstract Cubane derivatives with a given chemical formula and a given symmetry are enumerated by applying the elementary-superposition method (S. Fujita, Theor. Chim. Acta1992, 82, 473–498) of the unit-subduced-cycle-index (USCI) approach to a cubane skeleton of the point group Oh. A cycle index (CI) for a regular or an irregular case of chiral and achiral ligands is calculated in accord with such a given chemical formula. The CI is superposed elementarily onto respective subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs). Thereby, the numbers of fixed points (derivatives) are obtained with respect to the respective SCI-CFs to give a fixed-point vector (FPV). The resulting FPV is multiplied by an inverse matrix of the mark table of Oh to give an isomer-counting vector (ICV), which contains the numbers of 3D-structural isomers in an itemized fashion with respect to point-group symmetries. The concept of prochirality is examined by using cubane derivatives enumerated by the ICV.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Partial-Cycle-Index Method of the USCI Approach

Abstract By placing chiral and achiral proligands on the eight positions of a cubane skeleton of Oh, resulting cubane derivatives are counted as three-dimensional (3D) structural isomers by the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, 1991; S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers, University of Kragujevac, Kragujevac, 2007), where the numbers of 3D structural isomers with given constitutional formulas are itemized with the subsymmetries of Oh. For this purpose, partial cycle indices with chirality fittingness (PCI-CFs) for respective subgroups of Oh are calculated by using unit subduced cycle indices with chirality fittingness (USCI-CFs) and an inverse matrix of the mark table of Oh. After introducing ligand-inventory functions into the PCI-CFs, expansions of the resulting equations provide us with generating functions for giving the numbers of 3D structural isomers. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation for listing the resulting data in tabular forms.

Importance of the proligand-promolecule model in stereochemistry. I. The unit-subduced-cycle-index (USCI) approach to geometric features of prismane derivatives

Among the four methods of the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and Combinatorial Enumeration in Chemistry. Springer, Berlin, Heidelberg, 1991), the fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method have been applied to the combinatorial enumeration of prismane derivatives. These enumeration processes are based on the proligand-promolecule model, which enables us to take account of achiral and chiral proligands. Prochirality in a geometric meaning has been discussed in general by emphasizing the presence of enantiospheric orbits in enumerated prismane derivatives. An enantiospheric orbit accommodating chiral proligands (along with achiral ones) has been shown to exhibit prochirality by using various prismane derivatives as examples. On the other hand, the scope of pseudoasymmetry has been extended to cover such a rigid skeleton as prismane in addition to a usual pseudoasymmetric center as a single atom, where the proligand-promolecule model plays an essential role.

The Restricted-Subduced-Cycle-Index (RSCI) Method for Counting Matchings of Graphs and Its Application to Z-Counting Polynomials and the Hosoya Index as Well as to Matching Polynomials

Abstract The restricted-subduced-cycle-index (RSCI) method for generating Z-counting polynomials and the Hosoya indices (Z-indices) as well as matching polynomials has been developed by starting from subduced cycle indices (SCI) defined in the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, 1991). In the RSCI method, k-matchings of a given skeleton (or graph) for deriving these matters are regarded as restricted structures in which vertex substitution and edge substitution occur concurrently under a restricted condition that occupation of a common vertex does not occur. For the purpose of counting such restricted structures, the concepts of territory indicators and territory discriminants are introduced. Thereby, an RSCI for the skeleton (or graph) is derived from an SCI by the subduction to C1 (nonsymmetry). The RSCI gives the generating function for counting the numbers of restricted structures, which is further converted into a Z-counting polynomial, the Hosoya indices, as well as a matching polynomial. The versatility of the RSCI method is illustrated by applying to benzene, naphthalene, dodecahedron, and fullerene C60.

The Restricted-Subduced-Cycle-Index (RSCI) Method for Symmetry-Itemized Enumeration of Kekulé Structures and Its Application to Fullerene C60

Abstract The restricted-subduced-cycle-index (RSCI) method for symmetry-itemized enumeration of Kekulé structures has been developed as an extension of the unit-subduced-cycle-index (USCI) approach (S. Fujita, “Symmetry and Combinatorial Enumeration in Chemistry,” Springer-Verlag, 1991). In the RSCI method, such Kekulé structures are considered to be generated by edge substitution of a skeleton under a restricted condition that no selected edges have a common vertex. The restricted condition is formulated by means of the concept of territory indicators and territory discriminants, by which subduced cycle indices (SCIs) derived from unit subduced cycle indices (USCIs) are converted into restricted subduced cycle indices (RSCIs). The resulting RSCIs are used to evaluate marks (numbers of fixed points) in place of the SCIs used in the USCI approach, so that the numbers of Kekulé structures are calculated in a symmetry-itemized fashion. Moreover, the resulting data of enumeration are used to draw respective Kekulé structures by means of orbit indicators and edge indicators. The application of the RSCI method to fullerene C60 of Ih-symmetry is discussed in detail.

Enumeration of Three-Dimensional Structures Derived from a Dodecahedrane Skeleton under a Restriction Condition. II. The Restricted-Partial-Cycle-Index (RPCI) Method Based on Restricted Subduced Cycle Indices

The restricted-partial-cycle-index (RPCI) method for combinatorial enumeration under the restriction of no adjacency of ligands has been developed as a restricted version of the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry," Springer-Verlag (1991)). To take account of the restriction condition, (unrestricted) subduced cycle indices with chirality fittingness (SCI-CFs) of the USCI approach are converted into restricted subduced cycle indices with chirality fittingness (RSCI-CFs). Then, restricted partial cycle indices with chirality fittingness (RPCI-CFs) are derived from the RSCI-CFs, just as partial cycle indices with chirality fittingness (PCI-CFs) are derived from the SCI-CFs in the USCI approach. The resulting RPCI-CFs provide generating functions for restricted enumerations. The RPCI method using such RPCI-CFs is applied to enumeration of dodecahedrane derivatives under the restriction of no adjacency of ligands. Several enumerated derivatives are depicted and their symmetries are discussed to comprehend stereochemical properties such as pseudoasymmetry, sphericity, and prochirality.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Fixed-Point Matrix Method of the USCI Approach

Abstract Cubane derivatives with chiral and achiral proligands are counted as three-dimensional (3D) structural isomers by the fixed-point matrix method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, “Symmetry and Combinatorial Enumeration in Chemistry,” Springer-Verlag, 1991). The numbers of such 3D structural isomers are itemized with respect to their point-group symmetries, which are subgroups of the point group Oh for a cubane skeleton. For this purpose, the full list of unit subduced cycle indices with chirality fittingness (USCI-CFs) of Oh is constructed in a tabular form. Fixed-point vectors or fixed-point matrices, which are calculated by starting from USCI-CFs, are used to count 3D structural isomers in the form of isomer-counting matrices. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation.

The Fujita combinatorial enumeration for the non-rigid group of 2,4-dimethylbenzene

Using non-rigid group theory, it was previously shown that the full non-rigid group of 2,4-dimethylbenzene is an ummatured and isomorphic to the group C2?(C3wrC2) of order 36, where Cn is the cyclic group of order n, the symbols ? and wr stand for the direct and wreath products, respectively. Herein, it is first shown that this group has 12 dominant classes. Then, the Markaracter Table, the Table of all integer-valued characters and the unit subduced cycle index (USCI) Table of the full non-rigid group of 2,4-dimethylbenzene are successfully derived for the first time.

Group-theoretical Foundations for the Concept of Mandalas as Diagrammatical Expressions for Characterizing Symmetries of Stereoisomers

Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\)G and left coset representations (LCR) G(/H) for the group G and its subgroup H, a regular body of G-symmetry is defined as a diagrammatical expression for a right regular representation (C 1\)G, which is an extreme case of RCRs. The |G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of G and segmented into |G|/|H| of H-segments, which are governed by the RCR (H\)G. By regarding each H-segment as a substitution position, the H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating a molecule. The reference regular body (or H-segmented one) is operated by every symmetry operations of G to generate regular bodies (or H-segmented ones), which are placed on the vertices of a hypothetical regular body of G-symmetry. The resulting diagram (a nested regular body) is called a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the G-symmetry of a molecule. The effect of a K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the K-assemblage of the mandala (or the reduced mandala), where the resulting K-assemblies governed by the LCR G(/K) construct a |G|/|K|-membered orbit, which corresponds to a molecule of K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number of fixed points of K-assembled mandalas (or K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers. The relationship between a mandala and a reordered multiplication table is discussed.

Doubly-Colored Graphs as Graphical Models for Subductions of Coset Representations, Double Cosets, and Unit Subduced Cycle Indices

The concept of doubly-colored graphs is proposed to model subductions of coset representations, double cosets, and unit subduced cycle indices, which have been mathematically formulated in coset algebraic theory developed by Fujita (Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin, 1991).

組み合わせ化学のグラフ的モデル化

Traditional textbooks [1–3] on chemical group theory deal with linear representations and character tables of groups, which are usually directed to modeling molecular vibrations, infrared spectroscopy, and symmetrysimplification of secular-determinants in molecular orbital calculations. On the other hand, Fujita [4] adopted an alternative approach which is based on permutation representations (coset representations) and mark tablesg, where he developed the USCI (unit-subducedcycle-index) method [4, 5], aiming at studying combinatorial enumeration in chemistry. The two approaches of chemical group theory have been compared [6] after the development of the characteristic-monomial method for combinatorial enumeration on the basis of the former approach [7, 8]. While the former approach requires conjugacy relations between elements of a group only, the latter approach requires a knowledge of all group/subgroup relations which may not be trivial to find out. Usually a group G is expressed as the set of elements of symmetry g’s which it contains:

The unit-subduced-cycle-index methods and the characteristic-monomial method. Their relationship as group-theoretical tools for chemical combinatorics

Among the four methods of the unit-subduced-cycle-index (USCI) approach, the subduced-cycle-index (SCI) method and the partial-cycle-index (PCI) method have been discussed by using adamantane of Td-symmetry as a probe for enumeration problems, where USCIs are derived on the basis of permutaion representations, coset representations (CRs) and marks. After the examination of the SCIs and PCIs, Polya's theorem that is a standard method of chemical combinatorics has been derived from the USCI approach. As another approach, a new method called the characteristic-monomial (CM) method has been developed by virtue of charactereistic monomials (CMs). The CMs have been derived from Q-conjugacy representations and Q-conjugacy characters, which have been related to irreducible representations and irreducible characters of the standard repertoire of chemical group theory. The two approaches have been compared to discuss group-theoretical tools for chemical combinatorics on a common basis.