Cauchy-Frobenius

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

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... 

Thus, by an application of the Cauchy Frobenius Lemma 1.26, we obtain that the number of unlabeled m-multigraphs on n nodes is equal to... 

The basic concept is the following generalization of the Lemma of Cauchy Frobenius 1.26 that uses the notion of weight which is mostly obtained Irom the content ... 

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. 

First, 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). 

---