Subduced Cycle Indices Research Articles

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).

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.

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.

Enumeration of Three-Dimensional Structures Derived from a Dodecahedrane Skeleton under a Restriction Condition. I. The Restricted-Fixed-Point-Matrix (RFPM) Method Based on Restricted Subduced Cycle Indices

Enumeration of Three-Dimensional Structures Derived from a Dodecahedrane Skeleton under a Restriction Condition. I. The Restricted-Fixed-Point-Matrix (RFPM) Method Based on Restricted Subduced Cycle Indices

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.

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).

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.

Generalization of Partial Cycle Indices and Modified Bisected Mark Tables for Combinatorial Enumeration

Abstract Generalized partial cycle indices have been proposed as a generalization of partial cycle indices (PCIs) in order to enumerate isomers with a given set of subsymmetries. To evaluate the coefficient of each term appearing in such a generalized PCI, the natures of a mark table and of its inverse have been examined after the proposal of (modified) bisected mark tables and their inverses as well as of (modified) bisected tables of unit subduced cycle indices with chirality fittingness (USCI-CFs). The inverse of a (modified) bisected mark table and the (modified) bisected USCI-CF table have been applied to the enumeration of chiral isomers and achiral isomers.

Characteristic Monomial Tables for Enumeration of Achiral and Chiral Isomers

Abstract The characteristic monomial table of an achiral group is applied to combinatorial enumeration of the following types: (1) achiral isomers and enantiomeric pairs, (2) achiral isomers and chiral isomers, (3) enantiomeric pairs, and (4) achiral isomers. The cycle index of each case is obtained by using the same set of subduced cycle indices derived from characteristic monomials, where the coefficients of the subduced cycle indices are given in advance.

Pseudo-point groups and subsymmetry-itemized enumeration for characterizing the symmetries of tetrahydropyran and 1,3-dioxane derivatives

The pseudo-point group Ĉ2v is defined for characterizing the symmetries of tetrahydropyran and 1,3-dioxane. Their derivatives with a given formula and a given symmetry are enumerated by the unit subduced cycle index approach. Each derivative enumerated belongs to a subsymmetry of Ĉ2v so as to be classified into isoenergetic (Type I, I′ or II) or anisoenergetic (Type III or IV). The orbits in the enumerated derivative are discussed by the sphericity and chronality terms.

Unit subduced cycle indices for combinatorial enumeration

AbstractCombinatorial enumeration by means of unit subduced cycle indices (USCIs) is discussed by using the group I (A5) and the related groups as examples. A modified method for the derivation of USCIs is presented, where a subduced mark table is a key concept. Several properties of USCIs are discussed by clarifying the relationship between double cosets and the subduction of coset representations, which gives an alternative formulation of the USCIs. This treatment provides us with a new prospect concerning the properties of double cosets. The usefulness of subduced cycle indices and cycle indices derived from USCIs is exemplified by enumeration based on an icosahedron.

Enumeration of digraphs with a given automorphism group

Four methods are described for enumerating digraphs with a given automorphism group: (1) a generating-function method based on subduced cycle indices, (2) a generating-function method based on partial cycle indices, (3) a method based on the elementary superposition theorem, and (4) a method based on the partial superposition theorem. All of these methods are based on the concept of unit subduced cycle indices and construct a set of versatile tools for combinatorial enumeration. They are applied to the enumeration of five-vertex digraphs with a given automorphism group. The table of marks and its inverse for the symmetric group of degree 5 are recalled. The table of USCIs of this roup is obtained.

The USCI approach and elementary superposition for combinatorial enumeration

The elementary superposition theorems are presented for enumerating chemical compounds that contain achiral and chiral ligands. Subduced cycle indices (SCI-CF), partial cycle indices (PCI-CF), and cycle indices (CI-CF) with chirality fittingness are defined by starting from unit subduced cycle indices with chirality fittingness (USCI-CF). All of these indices afford generating functions that are proved to be applicable to combinatorial enumeration. In addition, the concept of elementary superposition with and without chirality fittingness is proposed to provide the elementary superposition theorems. These theorems provide us with a new methodology of enumerating compounds, in which the numbers of isomers are obtained without relying on generating functions and are itemized with respect to molecular formulas (weights) and symmetries. The ⊛ operation is defined on the basis of the elementary superposition. Thereby, we derive superposition theorems concerning the PCI-CFs and the CI-CFs. These are applicable to combinatorial enumeration.

Enumeration of organic reactions by counting substructures of imaginary transition structures. Importance of orbits governed by coset representations

Two methods for the enumeration of organic reactions are presented in order to take obligatory minimum valencies of a given skeleton into consideration. The first method is a generalization of Polya's theorem, in which the transitivity of the positions of the skeletons is explicitly considered. Thus, a permutation representation acting on the positions is reduced into cosec representations (CRs). In accord with this reduction, unit cycle indices derived from the CRs construct a generalized cycle index. The second method is based on the subduction of the coset representations. This contains useful concepts such as unit subduced cycle indices and subduced cycle index that afford a new type of generating functions.