A note on pólya enumeration theory

The power group method of de Bruijn and Harary in enumeration under group action of mappings between finite sets is extended to include correlation of group actions on domain and range. By relaxing the restriction of weight functions to be constant over orbits, more specific results concerning the enumeration of orbits by weight are obtained.This paper is dedicated to Professor Dr. Ernst Ruch on the occasion of this 65th birthday     

含μ_3－S准平面型四核钴簇合物的化学键性质

具有６４个价电子的含μ_３－Ｓ的平面型四核钴簇合物是偏离通常的电子计数规则。为探索其簇骼化学键特性，我们采用量子化学的ＣＮＤＯ／２－ＬＭＯ方法计算其电子结构。结果表明，μ_３－Ｓ分别与三个钴原子形成三个双中心键，近似平面型的四个钴原子的ｐ轨道形成两个ｐ－π共轭分子轨道，６４个价电子均处于成键轨道上。     

Subduction of coset representations

Summary Enumerations of compounds based on a parent skeleton with and without the influence of obligatory minimum valency (OMV) are reported. The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. This type of enumeration requires introduction of several new concepts that are derived from the subduction of coset representations, e.g., a unit subduced cycle index, a subduced cycle index and the number of suborbits.     

Subduction of coset representations. An application to enumeration of chemical structures with achiral and chiral ligands

Molecules derived from a parent skeleton are enumerated where both achiral ligands as well as chiral ligands are allowed. Chirality fittingness of an orbit is proposed in order to permit chiral ligands. The enumeration is conducted with and without consideration of obligatory minimum valency (OMV). The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. The importance of coset representations and their subduction by subgroups is discussed. The subduced representations are classified into three classes through their chirality fittingness, which determines the mode of substitution with chiral and achiral ligands. Several novel concepts such as a unit subduced cycle index and a subduced cycle index are given in general forms.     

Space fullerenes: computer search for new Frank–Kasper structures II

A Frank?CKasper structure is a 3-periodic tiling of the Euclidean space E ³ by tetrahedra such that the vertex figure of any vertex belongs to four specified fullerenes with, respectively, 12, 14, 15, and 16 faces. Frank?CKasper structures occur in the crystallography of metallic alloys, clathrates, zeolites, and in geometrical optimization. 27 such physical structures are known. In Dutour et ai. (Acta Crystallogr A 66:637?C639, 2010) we obtained, by computer enumeration, all 84 such structures with up to 20 cells in a reduced fundamental domain; 13 among them were known physical structures. In the present follow-up study, we managed, by improving the computation, to get all 37 new structures with 21 cells in a reduced fundamental domain. Those structures are described, using six invariants: group, the size of fundamental domain, mean coordination number $ \bar{f} $ , fraction sequence, cell orbits and major skeleton. We found pairs of distinct structures having all six invariants equal. So, we devised a new invariant, zigzag vector, and computed it for all 135 structures known from now; all have this invariant different. New bounds for $ \bar{f} $ and new directions (computational perspectives, number of Kekule structures, space octahedrites, space cubites) are also discussed.     

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.     

Symmetry specified enumeration of substituted derivatives: an easy solution to the complex problem

Several sophisticated methods to solution of symmetry specified enumeration problems are available in the modern literature. In this paper we propose a simple technique that allows one to manually compute the exact numbers of fixed-symmetry derivatives for a given structure either with inclusion or ignoring the substitution patterns. The basic idea of the method suggested consists in the derivation of Pólya-like cycle indices for the automorphism groups of specially constructed orbit partition graphs; the expansion of these indices and subsequent simple calculations result in the desired numbers of substituted derivatives with achiral substituents. Limitations of the new technique (and a method suggested earlier) depend on the relevance of the orbit partitions for particular subgroups of the point symmetry group. For illustration purposes, the results obtained for the prismane (D _3h ) and adamantane (T _d ) structures are discussed. In the former case the numbers of substituted derivatives can be found for all subgroups of the D _3h group, whereas in the latter case these numbers can be determined for eight out of eleven subgroups of the T _d point symmetry group. This work is based on the text of the lecture presented by the authors at the 5th All-Russia Conference on Molecular Modeling (Moscow, April 2007). The paper deals with the methodology and detailed treatment of applied aspects related to solution of enumeration problems for substituted derivatives with prescribed symmetry groups. Unlike the known methods of symmetry specified enumeration, the technique suggested is simple enough and may be regarded as generalization of the Pólya methodology, which is widely used by chemists. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 227–245, February, 2008.     

Analytical expressions for the count of LM-conjugated circuits of benzenoid hydrocarbons

A recursive method for enumeration of linearly independent and minimal conjugated circuits of benzenoid hydrocarbons had previously been given which is valid for several classes of benzenoid hydrocarbons. In the present article, the properties and constructions of unique minimal conjugated circuits and pairs of minimal conjugated circuits of a ring s in a benzenoid hydrocarbon B are investigated. An analytical expression for the count of LM-conjugated circuits of B is given which is based on the counts of Kekulé structures of selected subgraphs of B. By using the method, the LMC expression of any benzenoid hydrocarbon can be obtained. © 1996 John Wiley & Sons, Inc.     

On the interrelation between orbits and double cosets

Orbits and double cosets are intimately related: double cosets can always be looked upon as being orbits and often orbits can be identified with double cosets, in reverse. This note presents two such situations where orbits can be traced back to double cosets: the restriction of transitive permutation representations to subgroups and the cartesian product of two transitive permutation representations. These results readily apply to standard topics in chemical combinatorics dealing with isomers and isomerizations but equally like to less familiar combinatorial schemes such as Redfield's.     

Magnetoresistance effect,Runge–Lenz vector,and the torque vector due to anisotropy of electron orbits in cubic lattices submitted to the action of a constant magnetic field

Parameters due to an anisotropic character of the electron orbits in cubic metals submitted to the action of an external magnetic field are calculated. These parameters are, in the first step, a nonzero value of the metal magnetoresistance and—in the second step—the nonvanishing electron torque vector and the Runge–Lenz vector. Both these vectors depend on the electron angular momentum in the crystal lattices, which is also examined together with the radius of curvature of the electron orbits. The calculation of magnetoresistance is specified to the case of the body‐centered cubic lattice of metallic iron for which a comparison of the theory with the experimental data for magnetoresistance is also presented. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Generalization of the concepts of seniority number and ionicity

We present generalized versions of the concepts of seniority number and ionicity. These generalized numbers count respectively the partially occupied and fully occupied shells for any partition of the orbital space into shells. The Hermitian operators whose eigenspaces correspond to wave functions of definite generalized seniority or ionicity values are introduced. The generalized seniority numbers afford to establish refined hierarchies of configuration interaction spaces within those of fixed ordinary seniority. Such a hierarchy is illustrated on the buckminsterfullerene C\(_{60}\) molecule.     

Pseudo-point groups and chronality in enumeration of reaction pairs

Summary The pseudo-point group is constructed to characterize the symmetry of a basic pair of hexagonal reaction graphs having no par-bonds on its edges. Any pairs of reaction graphs (reaction pairs) are considered to be obtained by adding par-bonds to the edges of the basic pair; they are counted by the USCI (unit-subduced-cycle-index) approach. Thus, the six edges of the basic pair are assigned to the coset representation (/C _2v ). After the subduction of the is calculated, the partial-cycle-index method of the USCI approach is applied to the combinatorial enumeration of reactions pairs.Reaction pairs are classified to two categories, i.e. isoenergetic and anisoenergetic. An isoenergetic pair is concluded to be a self-reaction pair, while an anisoenergetic pair corresponds to a non-self-reaction pair. The concept of chronality is also discussed to clarify the symmetrical nature of the resulting orbits.     

Computer solutions for confidence intervals when the blank count time is an integer number of times greater than the sample count time

This paper extends past computer solutions for confidence intervals in paired counting. The net count is transformed to an integer; the resultant probability distribution is identical with that for detection limits when the blank is counted an integer number of times greater than the sample. Summation of the left tail of this distribution is used to determine left-hand and right-hand confidence limits with specified confidence levels for a given observed net count and known expected blank count in the sample count time. Poisson statistics are used throughout this paper and the Gaussian distribution is reserved for approximate results.     

烃同分异构体的计数

能否系统精确地解决高中化学学习中的同分异构体计数问题?本文在高中数学化学的背景知识之上,引入生成函数以及Pólya计数定理,在不考虑立体异构的前提下,从烷基的同分异构体计数入手,逐步推导烷烃、一烯烃、一炔烃苯的衍生物、二取代烷烃以及二烯烃的同分异构体计数的公式,并将计算结果与手工枚举进行对比,进一步证明其正确性。我们还归纳性地发掘了同分异构体计数的渐进特征并提出了以上烃类同分异构体计数的近似公式。对比国外的成果,本文尽力避免引入过多的数学概念,降低门槛,适合高中生阅读,加深对于烃结构的理解,培养计算思维;对比国内相同主题的研究,本文的推导形式清晰简洁,拓展性较强。     

Indexing scheme for classes of 𝒮N; partitions of N

The classes of the symmetric group 𝒮_N are identified by partitions of N. In this work an indexing scheme is presented which provides a dense enumeration of the classes of 𝒮_N. The method is based on a graphical representation of partitions of N, which also enables the determination of the class corresponding to a given number. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 421–426, 1997     

Expert system for elucidation of structures of organic compounds (ESESOC)

An algorithm for enumeration of stereoisomers due to asymmetric carbon, C = C double bond and so on has been developed. It consists of three steps. The output of stereoisomers can be represented by 2.5-dimensional connection table.     

Class-sum products in the symmetric group: Combinatorial interpretation of the reduced class coefficients

An algorithm for the evaluation of the structure constants in the class algebra of the symmetric group has recently been considered. The product of the class sum [(p)]_n that consists of a cycle of length p and n−p fixed points, with an arbitrary class sum in S_n, was found to be expressible in terms of a set of reduced class coefficients (RCCs), the p-RCCs. The combinatorial significance of the p-RCCs is elucidated, showing that they are related to a well-defined enumeration problem within S_p, which has to do with a certain refinement of the corresponding class multiplication problem. This is in contrast with the representation-theoretic evaluation of the p-RCCs, which requires the evaluation of products involving [(p)]_n for several values of n > p. The combinatorial interpretation of the p-RCCs allows the derivation of some of their previously conjectured properties and of some of the “elimination rules” that specific types of p-RCCs were found to satisfy. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 103–118, 1998     

A deterministic algorithm for constrained enumeration of transmembrane protein folds

A deterministic algorithm for enumeration of transmembrane protein folds is presented. Using a set of sparse pairwise atomic distance constraints (such as those obtained from chemical cross-linking, FRET, or dipolar EPR experiments), the algorithm performs an exhaustive search of secondary structure element packing conformations distributed throughout the entire conformational space. The end result is a set of distinct protein conformations, which can be scored and refined as part of a process designed for computational elucidation of transmembrane protein structures.     

Expert system for elucidation of structures of organic compounds (ESESOC)——Algorithm on stereoisomer generation

An algorithm for enumeration of stereoisomers due to asymmetric carbon, C=C double bond and so on has been developed. It consists of three steps. The output of stereoisomers can be represented by 2.5-dimensional connection table.