# Applications of Burnside's Formula

The applications of Burnside's formula in counting orbits has wide applications (I believe). But, whatever the books I followed on Group Theory, many (or almost all) of the applications mentioned in them are for "coloring problem" which involves roughly coloring vertices of a regular $n$-gon with different colors.

Q. What are the other simple applications of Burnsides theorem which I can present to undergraduates while teaching group theory?

• Counting graphs up to isomorphism (but don't ask me for details). – darij grinberg Jan 11 '15 at 14:32
• – Andreas Caranti Jan 11 '15 at 14:33
• Here's a list of fifty Burnside lemma and Pólya enumeration theorem examples from this site. – MJD Jan 11 '15 at 14:41
• Thanks very much for useful link @Caranti. – Groups Jan 11 '15 at 15:50
• @Groups, you're welcome! – Andreas Caranti Jan 11 '15 at 18:19

## 1 Answer

If you allow for Polya's enumeration theorem also here are a list of interesting problems that have been answered:

How many ways can you put 8 red, 6 green and 7 blue balls in 4 indistinguishable bins? (A sweet solution by Marko Riedel)

Using it to count graphs (An incredible solution by Marko Riedel)