Results of Enumeration

Listing of Results. A summary of the deduced algebraic expressions for the total numbers of isomers is given in Table 9. Specifications with relevance to symmetry are listed in the following. 

By passing we wish to point out a feature, which often is encountered in different kinds of enumeration the total numbers are the same as those of the unsymmetrical systems, but shifted one place on the list cf., e.g., the results on some polygonal systems representing polycydic conjugated hydrocarbons with arbitrary ring sizes (BrunvoU, Cyvin BN and Cyvin 1993a). The below diagram and table are supposed to elucidate the feature in question. 

The dots on the A axis are associated with the Mmirror-synunct cal ( 2 coronoids, while all the others pertain to the nnsymmetrical (C coronoids being Uin. nnmb. It holds in general that /for a given k is equal to 17for A + 1. Hence (cf. Table 9) 

Numerical values of the total number Hot k 20 are listed in Table 10. 

Exactly all the circular benzenoids 0 N S) with 5 18 can be perforated by a coronene hole to produce single coronoids 0(n 5). Then one has i T=n+6, 5=5 — 6. A detailed treatment on a complete enumeration of the isomers of the class 0 has been published (Cyvin BN, Brunvoll, Chen and Cyvin 1993). The smallest system of this class is kekulene. In the cited reference a coronoid with coronene hole(s) was referred to as coronenic . We quote a nice sentence therefrom A coronenic coronoid has a coronene corona hole. We shall not use the term coronenic in the following, although it is analogous with our term naphthalenic (Definition 3.3). A brief account on the title systems and their enumeration is given in the next two paragraphs. 

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The results of enumerations and classifications of polyhexes are reviewed and supplemented with new data. The numbers are collected in comprehensive tables and supplied with a thorough documentation from an extensive literature search. Numerous forms of the polyhexes are displayed, either as dualists or black silhouettes on the background of a hexagonal lattice. In the latter case, the Kekule structure counts for Kekulean systems are indicated. Emphasis is laid on the benzenoid systems (planar simply connected polyhexes). 

In Sect. 3 some classical results of enumeration of polyhexes are reported. They include Klamer numbers , Harary-Read numbers and Lunnon numbers , which were so beautifully (but not infallibly) sorted out for the first time by the Dusseldorf-Zagreb group [ 1 Reference to the Dusseldorf-Zagreb numbers is also contained in Sect. 3. Furthermore, original supplements to the different tables are found therein. 

Table 18 shows the results of enumeration of different SCS s [79]. Their forms for h up to 4 are depicted in Fig. 17. The dualist representation is employed. The horizon to the right of A is drawn as a straight line, indicating a forbidden region (on and below this line) for the SCS s. Similarly the forbidden region to the left of A is indicated by a straight line shifted one step up. 

In [95] the results of enumeration of interacting SAWs in two dimensions for iV < 30 and in three dimensions for A < 19 arc given. As a result of this study, they report a reentrant force-temperature diagram in three dimensions, but not in two dimensions, exactly as observed in the toy models in [96]. Based on much longer enumerations in two... 

In given work the possibilities enumerated above of varieties of thermal analysis used to reseai ch of solid solutions of hydrated diphosphates with diverse composition. So, for example, the results of differential-thermal analysis Zn Co j P O -SH O showed, that it steady in the time of heating on air to 333 K. A further rise of temperature in interval 333 - 725 K is accompanied with the masses loss, which takes place in two basic stages, registered on crooked TG by two clear degrees, attendant to removal 4,0 and 1,0 mole H O. On crooked DTA these stages dehydration registers by two endothermic effects. In interval 603 - 725 K on crooked DTA is observed an exothermal effect. 

The water analysis is incomplete unless the number of coliform bacteria present is determined as well. A multiple-tube fermentation technique can be used to enumerate positive presumptive, confirmed, and fecal coliform tests. Results of the tests are expressed in terms of the most probable number (MPN). That is, the count is based on a statistical analysis of sets of tubes in a series of serial dilutions. MPN is related to a sample volume of 100 ml. Thus, an MPN of 10 means 10 coliforms per 100 ml of water. 

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

It must be admitted that many chemical enumerations that have appeared in the literature are more in the nature of academic exercises than results of practical interest to the chemist. Thus, for example, no chemist really needs to know that the number of alkanes having 60 carbon atoms is 22,158,734,535,770,411,074,184 (see [PerD32]). However, the enumeration of compounds with a given frame can produce results of practical importance. 

Further progress in asymptotic tree enumeration was made by Otter, who in [OttR48] considered the problem of rooted and unrooted trees with maximum degree m. Having enumerated unrooted trees by the method already described, he proceeded to derive asymptotic estimates, and after ten pages of analysis arrived at a number of results, of which the following is typical ... 

As a second example, the chance of getting a one or two from throwing a fair die is j. What is the likely outcome if the fair die is tossed 10 times Enumeration could again be used to determine the probability of zero, one, two, etc., occurrences of one or two. As a result of the process the distribution of outcomes is shown in Figure 8.1 from which it is determined that... 

Table 8.1 Enumeration of the results of tossing a coin four times...

However, for the enumerations considered in the present paper no such complications arise and it is reasonable to rely on the results of extrapolations. The suggestion made by Flory and Fisk31 that the exact enumerations in the range n = 1 to 14 are dominated by immediate reversals is incorrect. The discussion of Section IV-G indicates that the nth step in the enumeration eliminates the effect of polygons of order n, and the contribution of the nth stage is of order l/nc. Thus the exact enumeration procedure is equivalent to assessing the asymptotic behavior of l/nc from the first 14 terms. 

Ozaki [1245] in his review classified perovskites prepared from metal alkoxides according to conditions of their crystallization. He described the following three possibilities of crystallization of perovskites (1) direct crystallization in the course of hydrolysis, (2) one-step process of thermal treatment of the amorphous hydrolysis products, and (3) crystallization as a result of the solid-state reactions between the first crystallized oxides. At present, it has become evident that the careful choice of processing conditions (which includes pre-hydrolysis and hydrolysis stages) allows most of the perovskites enumerated by Ozaki to be obtained without thermal treatment after hydrolysis. If the thermal treatment is, nevertheless, necessary, it is important to choose the appropriate atmosphere (air, oxygen, or oxygen-water vapor flow). 

In the previous examples for molecules with relatively few atoms, we studied and presented the results of all 31V-6 vibrations of the molecules. However, for molecules composed of a large number of atoms, often it is convenient to concentrate on a certain type of vibration. The prime example of this kind of investigation is the enumeration of the CO stretching modes in metal carbonyl complexes. 

Balaban published an early review [20] and recently a survey of enumeration results for benzenoids and coronoids [21] these two publications together clearly illustrate the development in the area. The most recent review so far with the most complete quotations of enumeration data for benzenoids and coronoids is found in a monograph by Gutman and Cyvin [22]. 

In general, this chapter does primarily not deal with methods and principles of enumerations, such as coding and nomenclature. Information on these issues are found in the different individual papers, which are cited, and especially in a recent review [31]. In the present treatise, emphasis is laid on the presentation of results in surveyable tables. The review of relevant literature results, which tends to be as complete as possible, is supplemented by a substantial amount of own results not published before. 

Survey of Enumeration Results for All Polyhexes 3.1 Benzenoids and Coronoids... 

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