Permutation groups

The Symmetry Properties of Wave Furictioris of Li3 Electronically Ground State in S3 Permutation Group... 

Symmetry Properties of TABLE XVTTI H3 Wave Functions in the < 3 Permutation Group ... 

Figure 2 40. To illustrate the isomorphism problem, phenylalanine is simplified to a core without representing the substituents. Then every core atom is numbered arbitrarily (first line). On this basis, the substituents of the molecule can be permuted without changing the constitution (second line). Each permutation can be represented through a permutation group (third line). Thus the first line of the mapping characterizes the numbering of the atoms before changing the numbering, and the second line characterizes the numbering afterwards. In the initial structure (/) the two lines are identical. Then, for example, the substituent number 6 takes the place of substituent number 4 in the second permutation (P2), when compared with the reference molecule.

In order to handle the stereochemistry by permutation groups, the molecule is separated at each stereocenter into a skeleton and its ligands (Figure 2-74). 

The preceding section gave a briefintroduction to the handling of the stereochemistry of molecules by permutation group descriptors. Here we discuss this topic in more detail. The treatment is largely based on ideas introduced in Ref. [100],... 

The stereochemistry of reactions has to be handled in any detailed modeling of chemical reactions. Section 2.7 showed how permutation group theory can be used to represent the stereochemistry of molecular structures. We will now extend this approach to handle the stereochemistry of reactions also [31]. 

Let us first repeat the essential features of handling the stereochemistry of molecular structures by permutation group theory ... 

Figure 3-23. The treatment ofthe stereochemistry ofa further S,y2 reaction by permutation group theory.

The stereochemistry of reactions can also be treated by permutation group theory for reactions that involve the transformation of an sp carbon atom center into an sp carbon atom center, as in additions to C=C bonds, in elimination reactions, or in eIcctrocycHc reactions such as the one shown in Figure 3-21. Details have been published 3l]. 

The stereochemistry of reactions can be treated by means of permutation group theory. 

Cayley s extensive computations have been checked and, where necessary, adjusted. Real progress has been achieved by two American chemists, Henze and Blair Not only did the two authors expand Cayley s computations, but they also improved the method and introduced more classes into the compound. Lunn and Senior , on the other hand, discovered independently of Cayley s problems that certain numbers of isomers are closely related to permutation groups. In the present paper, I will extend Cayley s problems in various ways, expose their relationship with the theory of permutation groups and with certain functional equations, and determine the asymptotic values of the numbers in question. The results are described in the next four chapters. More detailed summaries of these chapters are given below. Some of the results presented here in detail have been outlined before ... 

The combinatorial problem on permutation groups stands out for its generality and the simplicity of the solution. The following... 

Chapter 1 expands on the above introduced concept of "configurations which are equivalent with respect to a permutation group". General rules are established and some related topics are mentioned. 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

The definitions concerning figures will be followed by those on permutation groups. The terminology is suggestive. Symbols will have the same meaning throughout. 

Permutation groups. Consider a permutation group H of order h and degree s. 

We now have to establish the relationship between the permutation group H and the figure collection [ ]. 

The problem of which the example in Sec. 2 represents a very special case can be stated as follows Given the collection of figures [ ], the permutation group H and the content (k,H,m), determine the number of nonequivalent configurations of content (k,Jl,m) with... 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

In the sequel we will see that the proposition holds for an arbitrary permutation group and we will refer to it as the theorem or the main theorem. 

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