Counting series

One of the attractions of Polya s paper is the elegant way that generating functions are introduced and used. In order to summarize the information about the figures and their contents, Polya introduced the "figure generating function" (also frequently called the "figure counting series")... 

In order to ply Pdlya s theorem, we substitute for hydrogen-labelled benzene (corresponding to substitution of one or more hydrogen atoms by a halogen atom or a hydrogen isotope) the figure counting series ... 

If one now substitutes the ligure-counting series sj = 1 + one obtains the counting polynomial ... 

A first example is C/H-labelled ethene which for H-labelling has the configuration-counting series ... 

By substituting two figure counting series s=l-hA into this... 

By substituting appropriate figure-counting series, one can calculate the number of isomers for apically (t = 1 + a), equatorially (s = 1 + e) or centrally (u = 1 -F c) labelled isomers. 

In this subsection, we briefly summarize results relevant to labeled graphs. We then survey the work on counting series by Cayley, Polya, Ffarary,... 

Theorem (Polya). The configuration generating function, or counting series, C(x), is obtained by substituting the figure generating function, c(x), in the cycle index, by replacing every occurrence of Sk in the cycle index by c x ). Thus,... 

As an illustration, the generating function c(x) = - - substituted in the benzene cycle index of Eq. [5] gives the counting series ... 

Table 2 Cycle Indices and Counting Series for Some Hydrocarbons Cages...

Primary alcohols. The primary alcohols are of the form R-CH2-OH, where R is an alkyl group with n-1 carbons atoms. To maintain the correct number of carbon atoms, the counting series for primary alcohols becomes... 

Tertiary alcohols. The formula for a tertiary alcohol is HO-C(Ri)-(R2HR3). The counting series is obtained with the same arguments as for secondary alcohols but with the permutation group S3 instead of S2 ... 

Aldehydes and ketones. These compounds have the form Ri-C=0—R2, where Ri and R2 are alkyl groups and, possibly, hydrogen atoms. As hydrogen atoms are included, the counting series is... 

Alkynes. The formula for acetylene compounds takes the form Ri-C=C—R2, and because two additional carbon atoms are available when no terminal hydrogens exists, the counting series is... 

Esters. The general ester formula is Ri-C=0—OR2, with Rj and R2 being alkyl groups. Ri can be a hydrogen atom but not R2 otherwise the compound would be an acid. Consequently, the counting series is A(x) for Rj and A x) — 1 for R2. One more carbon must be added in the ester formula. The final counting series becomes... 

We now turn our attention to a class of compounds that is not of primary importance in chemistry, but the results are used later to derive the counting series for alkanes. These structures are of the type R1-R2, where Ri and R2 are non-hydrogen alkyl groups with counting series A(x) — 1. The permutation group is S2, and the counting series is... 

Hydroxyl ethers. In a recent development, Wang, Li and Wang proposed a counting series for compounds of the form C,H2,+20/j which is a first step toward a general counting series for molecular formulas. Their technique involves Polya s cycle index and two generating functions for alkyl groups... 

Now that we have determined the permutation group we can count stereoisomers using the formulas obtained with structural isomers but by replacing the group S3 by C3. For instance, from Eq. [10], the counting series for alkyl groups becomes... 

All isomer counts we have given so far are derived from Polya s theorem and the alkyl group counting series. Our intention was to illustrate the power of Polya s counting theory and to make things easier to follow because all... 

Table 3 Counting Series for the Stereoisomers of Functionalized Alkanes ...

The X on the right-hand side come from the fact that l/j = 1, whereas Si=Di = 0. Substituting Eq. [21] into Eq. [22] we derive the counting series for bond-rooted catafusene hydrocarbons ... 

The derivation of the counting series for unlabeled catafusenes can be found in Harary and Read." Their solution makes use of Otter s formula " the same way the counting series for alkanes was derived with counting series for labeled alkanes and alkyl groups. The counting series for unlabeled catafusenes is... 

So far we have regarded a catafusene and its mirror image as distinct, provided that the catafusene has no symmetry that would allow it to be rotated into its mirror image. The counting series in Eq. [25] was corrected by Harary and Read" to count only once catafusenes and their mirror images. 

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