A good example is self-complementary graphs. It's not hard to count unlabeled self-complementary graphs using Burnside's lemma, but there seems to be no reasonable formula for counting labeled self-complementary graphs.

A good example is self-complementary graphs. It's not hard to count unlabeled self-complementary graphs using Burnside's lemma, but there seems to be no reasonable formula for counting labeled self-complementary graphs.

Unlabeled self-complementary graphs are counted by sequence A000171 in the OEIS. A paper on counting labeled self-complementary graphs is Shinsei Tazawa, A new counting methods, including the issue of counting labelled self-complementary graphs, arXiv:0909.2314 [math.CO]. This paper computes the number of labeled self-complementary graphs on $n$ vertices for $n$ up to 9.

A good example is self-complementary graphs. It's not hard to count unlabeled self-complementary graphs using Burnside's lemma, but there seems to be no reasonable formula for counting labeled self-complementary graphs. Note that 182830 asks the same question.

A good example is self-complementary graphs. It's not hard to count unlabeled self-complementary graphs using Burnside's lemma, but there seems to be no reasonable formula for counting labeled self-complementary graphs.