Combinatorial Formulas

Hosoya (1986a) observed a resemblance between the polynomial in (1) and the characteristic polynomial for benzene, viz. a — 6x4 gx — 4. He writes The reason for this mystic coincidence, however, is not known. On the other hand, a substantial amount of work has been done in the area of combinatorial K formulas in the form of polynomials for primitive coronoids in general, and especially for those of regular hexagonal symmetry as in the case above 

This formula was derived by Bergan. A determinant form, produced by Hosoya, was communicated by Cyvin SJ, Brunvoll, Cyvin, Bergan and Brendsdal (1991). An interesting general formulation for the Kekule structure counts of a primitive coronoid with specified lengths of segments reads (Cyvin SJ 1990 Cyvin SJ, Cyvin, Brunvoll, Hosoya et al. 1991) 

A combinatorial K formula of a different kind, viz. in terms of exponential functions, has been deduced for the class of primitive coronoids vrith S equidistant segments (Balaban, BrunvoU and Cyvin 1991 Vol. 1-6.4). If the length of each segment is a + 1, then (Bergan, Cyvin BN and Cyvin 1987) 

The case a = 1 is also interesting, especially because Fibonacci numbers and Lucas numbers turn up (Bergan, Cyvin BN and Cyvin 1987 Cyvin SJ 1990)  

Here the Fibonacci and Lucas numbers are defined by and Lq = 1, = 3, respectively. 

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In this expression, the degeneracy factor gN represents the number of ways the TV molecules may be placed on M sites, and is given by the combinatorial formula (see Equation (3.52) in Section 3.4a.2 or Equation (2.46) in Section 2.6a) ... 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

In the first three regions the expressions for (rl, r2, R) are simple combinatorial formulas containing lA, lB, and m times a product of powers of ru r2, and R, while in the last region the expression contains a (finite) sum of powers of ru r2, and R ... 

Why Enumerate . - Clearly enumeration has played an important role in the history of chemistry. But does it still Are the noted enumerations just historical anachronisms Is enumeration irrelevant for modem interests in quantitative descriptions of different substances Indeed in all the areas we have noted, one may indeed argue that enumeration is but a first step towards a more comprehensive characterization and undertaking. Combinatorial formulae often merely identify two different enumerations to have equal values, with one of the enumerations being the easier to perform. We may note for instance that isomer enumeration in Polya theory identifies this enumeration to that of the enumeration of certain equivalence classes of functions. With the counts for two different sets of objects being equal, there often is a natural bijection i.e. a one-to-one correspondence) between the two sets, so that the objects of one set may be used to represent (or even name) those of the other. Thence for the case of chemical isomers again, the mathematical set of objects offers a nomenclature for the isomers. Conversely too, granted a nomenclature, a possibility for enumeration is offered one seeks to enumerate the names (which presumably exhibit some systematic structure). In some sense then a sensible nomenclature and enumera-... 

Each of the n electrons of the 4f configuration is associated with one of the seven 4f wavefunctions and may have a spin of Vi. There are a number of ways of associating the n electrons with the 4f orbitals, taking the spin into consideration this number corresponds to the multiplicity (or degeneracy) of the configuration and is given by the following combinatorial formula ... 

Now it is relatively easy to find a combinatorial formula for the number of parallel edges, which is the same in all the three directions for one of the systems under consideration. It was found... 

A circular single coronoid, 0, (Par. 4.6.2) is a circular benzenoid perforated by a naphthalene hole (cf. also Par 5.5.3). Cyvin SJ (1991b) presented a complete mathematical solution in terms of combinatorial formulas for the numbers of 0 isomers. 

The variation of polymerization constants with temperature is known for the system FeO - Si02 (Distin et al., 1971 Masson, 1972). The proportions of different structons can thus be cal-ciilated at different temperatures using the combinatorial formulae in Table 1. Activities of fayalite in the melt can therefore be calciilated from the expression ... 

A combinatorial formula relating intersections and unions of a collection of overlapping sets. 

In practice, the constrained case plays really an important technical role that will soon become clear. And exactly connected to this constraint we see why we do not allow working with the simple random walk q = 0) we would have to require N 2N. Of course this is not much of a problem, but we decide to privilege simpler notations. Possibly the reader is familiar with the elementary combinatorial formulas available for the simple random walk [Feller (1966), Chapter III] and those formulas are of course crucial for the intuition on random walks but results that are, for all practical purposes, as precise are available for the (p, g)-walks (and beyond). A part of these results, the ones that we need, are recalled in Appendix A.6. 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

The method to derive T was given by Cayley, who established the functional equation in the form of (1) for the function t(x). Cayley s computations of R are more laborious. Henze and Blair have derived the recursion formula (2.56) by direct combinatorial considerations, without knowledge of the functional equation. Here, (2.56) is a consequence of the functional equation (4). 

Polya 3, 4, 5. The last paper contains a direct proof for the equivalence of the recursion formulas based on combinatorial considerations and on the functional equation (4). 

Assume that the two sites are separated by N steps in the x-direction. (The formula can be straightforwardly generalized to arbitrary locations of the two trap sites.) The number of extra kinks in the path of length N > N is K, which is the number of pairs of non-essential steps. The formula given then follows from the combinatorials of N total steps which may be taken in any order, and which consist of six classes of objects, N + k, forward steps (in the -I- x direction), k, backward steps ( — x), ky sideways steps ( 4- y), ky sideways steps back ( — y), and k ( + z), and kj ( — z), where we must also have that k, H- ky 4- k = K. Asymptotically for large N > N the number of paths grows as 6 /N ... 

The map can be finite or infinite and some holes can be i-gons with i e R. If R = r, then the above definition corresponds to (r, q)gen-polycycles. If an (R, q)gen-polycycle is simply connected, then we call it an (R, )-polycycle those polycycles can be drawn on the plane with the holes being exterior faces. (R, )-polycycles with R = r are exactly the (r, interior faces is that polycyclic hydrocarbons in Chemistry have a molecular formula, which can modeled on such polycycles, see Figure 7.1. The definition of (R, < )-polycycles given here is combinatorial we no longer have the cell-homomorphism into r, q). We will define later on elliptic,... 

One other problem is combinatorially large search spaces. There are 100 million potential candidates for rubber components formulation that are possible. Some other examples we have worked at Purdue have involved 1020 to 1030 different molecules. Another issue is that typically there are limited and uncertain data. Most often, combinatorial chemistry approaches do not succeed in these cases because obtaining the results is time and labor intensive. Fuel additives design, for example, requires dismantling the engine for every test of a new formula. 

As a final example it should be mentioned that precondensed enones, prepared by standard Knoevenagel condensation of the aldehyde with the CH-acidic carbonyl component, when reacted with thioureas provided 1,3-thiazines 37, which are isomeric to thio-Biginelli compounds of the general formula 14 (see Figure 4.5). A published report describes the combinatorial synthesis of a library of 29 derivatives of thiazines 37 utilizing polymer-supported reagents and catalysts [168]. 

Many researchers tried to explain the secret of the Clar s aromatic sextet theory, or hypothesis from quantum-chemical points of view. However, those trials have been failing until the graph and combinatorial theories came to be applied to this challenging problem [9,10]. In the following discussion it will be shown how various techniques and concepts of the graph theory are useful for realizing and formulating not only the fantastic theory of Clar but also the mathematical beauty of the structural formula of aromatic hydrocarbons. 

Outline a design for a combinatorial synthesis for the formation of a combinatorial library of nine compounds with the general formula B using the Furka mix and split method. Outline any essential practical details. Details of the chemistry of peptide link formation are not required it is sufficient to say that it is formed. 

Statistical methods are most widespread. Macromolecules with and without ring formation can be described by methods of combinatorial algebra [8]. Processes of polymerization (and simplest cases of destruction) are assumed to be random, that allows the derivation of the formulas for the number of JT-mers and the calculation of the weight fractions, the distribution functions, and the averaged molecular weight [8]. However, the problem of the general description of destruction is not still solved. 

Explicit formulas for the numbers of the title systems were developed, in a simple combinatorial way, by Balaban and Harary [13] cf. also Balaban [58], The derivation [13] involves a treatment of the subclasses of unbranched catafusenes with specific symmetries. It is outlined in the following, basically in the version of Brunvoll et al. [59] (cf. also Balaban et al. [60]), and is supported by illustrations. 

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