Polya counting theory

CraT80 Craven, T. C. An application of Polya s theory of counting to an enumeration problem arising in quadratic form theory. J. Combinatorial Theory A 29 (1980) 174-181. 

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... 

The first chemical application of graph theory dates back to 1875 when William Clifford proposed the solution for the counting of alkane isomers. The modern chemical graph theory started with the works of Henze and Blair in 1931 [Henze and Blair, 1931a, 1931b, 1933,1934] and Polya in 1936 [Polya, 1936, 1937a, 1937b Polya and Read, 1987]. 

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-... 

The isomer counts are in this case readily obtainable even without knowledge of Polya theory, but the method extends to rather much more comphcated skeletons. If one were to wish not to distinguish cis and trans structures, then the group would be extended to A U (12)A-... 

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