Counting unlabeled structures

we need a formula for the number of orbits. The following result originated from A. Cauchy and G. Frobenius (19th century) and is sometimes erroneously attributed to Burnside [220,345] who in fact proved an even stronger result (see below). 

26 Lemma (Cauchy-Frobenius, on the number of orbits) Consider a finite action 

the number of orbits ofG on X is the average number of fixed points  

27 Example (Naphthalene, cont.) In Example 1.7, the sets of fixed points of the four permutations of naphthalene are 

The Lemma of Cauchy-Frobenius is basic and very important. It holds since there is an interesting connection between the orbit G(x) e X and the stabilizer 

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The preceding discussion has shown how unlabeled structures, described as orbits of finite groups on finite sets, can be counted. What is still missing is a construction of unlabeled structures. In order to prepare it, we introduce further notions and examples of group actions ... 

Besides counting orbits by weight there is another refinement, the enumeration by symmetry. For certain applications it is important to evaluate the number of unlabeled asymmetric structures. Our approach uses a theorem from Burnside, which was mentioned briefly above. It is much stronger than the Lemma of Cauchy-Frobenius. 

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