 Unlabelled combinatorial structure

An unlabelled combinatorial structure is a combinatorial structure considered up to an appropriate isomorphism, so the names assigned to underlying elements do not distinguish objects. For graphs, this means a bijection of vertices preserving adjacency. A different drawing or vertex numbering can therefore represent the same unlabelled graph.

On three labelled vertices there are $2^{\left(\genfrac{}{}{0ex}{}{3}{2}\right)}=8$ simple graphs, but only four unlabelled types: the empty graph, one edge with an isolated vertex, the two-edge path, and the triangle. Simply dividing eight by $3!$ cannot work because these types have different numbers of automorphisms.

More generally, relabelling gives a group action of the symmetric group on labelled structures. Unlabelled structures are its orbits, and the orbit-stabilizer theorem explains how automorphisms affect orbit size. Burnside's lemma counts the orbits by averaging fixed-point counts.