Graphs to chemical structures 2. Extended sphericity indices of cycles for stereochemical extension of Pólya’s coronas

In order to enumerate nonrigid isomers, we have proposed the proligand approach, where extended sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. Then, cycle indices with chirality fittingness have been defined so as to enumerate nonrigid stereoisomers with chiral and achiral ligands. Results of the proligand approach using tetramethylmethane as an example have been compared with those based on Pólyas corona. Thereby, Pólyas corona is concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly.     

Sphericities of Double Cosets,Double Coset Representations,and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a _d , c _d , or b _d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.     

Graphs to chemical structures 1. Sphericity indices of cycles for stereochemical extension of Pólya’s theorem

Pólyas theorem has been concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly. In order to take account of chiral ligands along with achiral ones, coset representations (CRs) for cyclic subgroups have been examined to classify permutations of the CRs into proper and improper elements. As a result, a k-cycle contained in each permutation has been classified into an enantiospheric, homospheric, or hemispheric one. Thereby, sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. On the basis of the sphericity indices, cycle indices with chirality fittingness (CIs-CF) have been defined in place of Pólyas cycle indices. The CIs-CF have been proved to be capable of enumerating of stereoisomers with chiral and achiral ligands. Their capabilities have been confirmed by using allene derivatives as examples.     

Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes

Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita’s proligand method formulated in Parts 1–3 of this series [Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously [Fujita in Tetrahedron 47:31–46, 1991], a planted promolecule is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands. Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating functions are based on cycle indices with chirality fittingness (CI-CFs), which are 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. For the purpose of evaluating c _d recursively, the concept of diploid is proposed, where the nested nature of c _d is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x ²), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem initiated by mathematician Cayley [Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

The combinatorics of cation-deficient close-packed structures

Pólya's enumeration theorem is used to derive an algorithm for counting all hexagonal close-packed structures with a unit cell of given size, having composition MX₂, where X is an hcp anion, M is an octahedrally coordinated cation, and no face sharing is permitted between octahedra. Generalizations of this algorithm to enumerate ordered derivatives of these structures, hcp structures with tetrahedral instead of octahedral cations, and similar structures having different stacking sequences among the close-packed layers are sketched.     

Substitution symmetry

Pólya's enumeration theory and its generalizations are refined to count derivatives of symmetrical parent compounds with any specified subsymmetry. Equivalently, enumeration of orbits of mappings, upon which a group acts by acting on their domain and their range, is refined to count orbits with stabilizers in any specified conjugacy class of subgroups.     

Graphs to chemical structures 3. General theorems with the use of different sets of sphericity indices for combinatorial enumeration of nonrigid stereoisomers

The extended sphericity indices of k-cycles, which were defined in Part 2 of this series (S. Fujita, Theor Chem Acc, Online: http://www.springerlink.com/index/10.1007/s00214-004-0606-z) according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle, are further extended to prove more general theorems for enumerating nonrigid stereoisomers with rotatable ligands. One of the extended points is the use of different sets of sphericity indices to treat one or more orbits contained in skeletons and ligands. Another is to take account of chirality in proligands and sub-proligands, the latter of which are introduced to consider further inner structures of ligands. Two theorems for enumerating nonrigid stereoisomers are proved by adopting two schemes of their derivation, i.e., the scheme ``positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand ⇐ sub-proligands)' and the scheme ``(positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand)) ⇐ sub-proligands'. The theorems are applied to the stereoisomerism of trihydroxyglutaric acids. Thereby, it is demonstrated where Pólya's theorem and other previous methods are deficient, when applied to the enumeration of stereoisomers.     

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 T _d -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, Pólya'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.     

Enumerations of chemical structures by subduction of coset representations. Correlation of unit subduced cycle indices to Pólya's cycle indices

Subduction of the coset representations and the related concepts such as unit subduced cycle indices and subduced cycle indices yields the foundation for a new type of generating function for enumerating chemical structures. This method is related to Pó1ya's theorem.     

The USCI approach and elementary superposition for combinatorial enumeration

Summary 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.     

Enumerating treelike chemical graphs with given path frequency

The enumeration of chemical graphs satisfying given constraints is one of the fundamental problems in chemoinformatics. In this paper, we consider the problem of enumerating (i.e., listing) all treelike chemical graphs from a given path frequency. We propose an exact algorithm for enumerating all solutions to this problem on the basis of the branch-and-bound method. To further improve the efficiency of the enumeration, we introduce a new variant of the compound enumeration problem by adding a specification on the number of multiple bonds to the input and design another exact enumeration algorithm. The experimental results show that our algorithms can efficiently solve instances with larger sizes that are impossible to solve by the previous methods. In particular, we apply the latter algorithm to the enumeration problem of the special treelike chemical structures-alkane isomers. The theoretical and experimental results show that our algorithm works at least as fast as the state-of-the-art algorithms specially designed for generating alkane isomers, however using much less memory space.     

Graphs to chemical structures 5. Combinatorial enumeration of centroidal and bicentroidal three-dimensional trees as stereochemical models of alkanes

Three-dimensional trees (3D-trees), which are defined as a 3D version of trees, are enumerated by Fujita’s proligand method formulated in Part 1 to Part 3 of this series (Fujita in Theor Chem Acc 113:73–79, 113:80–86, 2005; 115:37–53, 2006). Such 3D-trees are classified into centroidal and bicentroidal 3D-trees, which correspond to respective promolecules having proligands as substituents. In order to enumerate such centroidal and bicentroidal 3D-trees, cycle indices with chirality fittingness (CI-CFs) are formulated as being composed of three kinds of sphericity indices, i.e., a _d for homospheric cycles, c _d for enantiospheric cycles, and b _d for hemispheric cycles. The CI–CFs are capable of giving itemized results with respect to chiral and achiral 3D-trees so that they are applied to derive functional equations (a(x), c(x ²), and b(x)). The generating functions of planted 3D-trees, which are formulated and calculated elsewhere, are introduced into such functional equations. Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated and collected up to a carbon content of 20 in a tabular form. Now, the enumeration problem initiated by a mathematician Cayley (Philos Mag 47(4):444–446, 1874) has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

Comments on McBride's completion of Kroner's proof that hydrogens of benzene are homotopic

This paper shows that McBride's completion of Kroner's proof (J. M. McBride, J. Am. Chem. Soc.102, 4134 (1980)) can be immediately obtained using the methods developed by the author in the context of isomer enumeration and NMR-signal enumeration with a theorem of Pólya. The method is illustrated with both rigid and non-rigid molecules.     

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 Pólya'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.     

Systematic enumeration of nonrigid isomers with given ligand symmetries

A new method for enumerating nonrigid isomers with rotatable ligands has been developed so as to take the symmetries of the ligands into consideration. The method has been based on extended partial cycle indices and has been applied to the enumeration of methyl ether derivatives, tetramethylallene derivatives, and 2,2-dimethylpropane derivatives. These results have been compared with the enumeration results of the corresponding promolecules. The factorization of terms in generating functions has been discussed so that the new method is capable of examining the relationship between promolecules and molecules quantitatively.     

The enumeration of electron-rich and electron-poor polyhedral clusters

We present as dual processes the capping of closed triangulated polyhedra with apical atoms and the making of holes in such polyhedra either by puncture of the surface or by excision of atoms and their edges. These processes are shown to generate stable chemical species containing respectively less or more than 2n + 2 skeletal electrons. The former species are designated as electron-poor whereas the latter are called electron-rich. Pólya's enumeration method is used to enumerate the distinct ways of capping and excising the closed, triangulated polyhedra to yield systems containing from four to twelve vertices. For the enumeration of cappings the appropriate cycle index is that of the dual of the polyhedron being capped, whilst for the enumeration of the excisions the cycle index is that of the polyhedron being excised.     

The numbers of chiral and achiral alkanes and monosubstituted alkanes

Whereas the theory for the enumeration of the optical isomers of the lakyl radicals and the alkanes has long been understood, this is not the case for the corresponding archiral isomers. We present for the first time recurrence formulae for counting the number of archiral isomers of the alkyl radicals and the alkanes. For chiral and archiral alkanes and monosubstituted alkanes, numerical results up to C₁₄ are tabulated.After presenting the history of the problem and the necessary definitions, we proceed to derive functional equations on the various generating functions, which readily yield the more explicit recurrence formulae usefule for numerical calculations. In the process, we first re-derive Pólya's expression for planted steric trees using his classical enumeration theorem. This result is then extended to the enumeration of free steric trees using the now standard tree-counting method due to Otter and known as a dissimilarity characteristic equation.By definition, a steric tree is a quartic tree (all points having degree 1 or 4) in which the four neighbors of every carbon point are given a tetrahedral configuration. Building on the methods of the first two authors for counting chiral and archiral trees in the plane, we obtain the formula for counting achiral steric trees, thus setting a problem first enunciated by van't Hoff and Le Bel in 1874.     

Combinatorial enumeration of nonrigid isomers with given ligand symmetries on the basis of promolecules with a subsymmetry of D infinity h

To enumerate nonrigid isomers with given ligand symmetries on the basis of a D infinity h skeleton, the concept of extended partial cycle indices (extended PCIs) proposed newly has been combined with the concept of promolecules proposed previously. The infinite nature of the D infinity h-group is concealed by adopting the factor group of finite order, D infinity h/C infinity (= K). Thus, the partial cycle indices with chirality fittingness (PCI-CFs) for the factor group K are calculated and combined with the PCIs for ligand symmetries so as to give the extended PCIs for various itemized enumurations. This method has been successfully applied to the enumeration of ethane derivatives, where the full enumeration based on K has been compared with partial enumerations based on K as well as with those based on the factor group C infinity h/C infinity (= K3 [symbol: see text] K). Each term of the resulting generating functions has been factorized into a pair of factors to represent ligand constitutions. Thereby, the depiction of resulting molecules can be conducted systematically so as to provide the maps of ethane derivatives corresponding to all of the substitution types.     

Combinatorial approach to group hierarchy for stereoskeletons of ligancy 4

Fujita’s proligand method developed originally for combinatorial enumeration under point groups (Fujita in Theor Chem Acc 113:73–79, 2005) is extended to meet the group hierarchy, which stems from the stereoisogram approach for integrating geometric aspects and stereoisomerism in stereochemistry (Fujita in J Org Chem 69:3158–3165, 2004). Thereby, it becomes applicable to enumeration under respective levels of the group hierarchy. Combinatorial enumerations are conducted to count inequivalent pairs of (self-)enantiomers under a point group, inequivalent quadruplets of RS-stereoisomers under an RS-stereoisomeric group, inequivalent sets of stereoisomers under a stereoisomeric group, and inequivalent sets of isoskeletomers under an isoskeletal group. In these enumerations, stereoskeletons of ligancy 4 are used as examples, i.e., a tetrahedral skeleton, an allene skeleton, an ethylene skeleton, an oxirane skeleton, a square planar skeleton, and a square pyramidal skeleton. Two kinds of compositions are used for the purpose of representing molecular formulas in an abstract fashion, that is to say, the compositions for differentiating proligands having opposite chirality senses and the compositions for equalizing proligands having opposite chirality senses. Thereby, the classifications of isomers are accomplished in a systematic fashion.     

Calcul de chiralité quantitative par la méthode des moindres carrés

It is shown how the root mean square chiral index is computed without enumerating all the permutations of a set of n atoms. As an example, the quantitative measure of chirality is achieved for various helicene derivatives. It is also shown that the chiral index of a protein backbone is computable without enumeration.