Number of unlabeled hypergraphs (A003180)

Question

I'm looking for the number of unlabeled hypergraphs on n nodes and stumbled upon the comments of A003180 in OEIS. Can somebody please explain to me how that sequence relates to the number of unlabeled hypergraphs?

For example, I think there should be 6 unlabeled hypergraphs on 2 nodes, the edge set could consist of:

• nothing
• 1 node
• 2 nodes
• nothing + 1 edge ("edge" meaning edge containing all two nodes)
• 1 node + 1 edge
• 2 nodes + 1 edge

But the sequence claims it should be 4.

Answer 1

Your enumeration is correct for OEIS sequence A000612 where hypergraphs have hyperedges of non-empty subsets. The Wikipedia article Hypergraphs states

There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.

The comment in A003180 assumes that empty hyperedges are allowed so A003180(n) = 2*A00612(n). I have added an extra comment to clarify this issue. Note that the OFFSET of A003180 is $0$ so A003180(2) = 12 is the number of hypergraphs with empty hyperedges allowed also.

Thanks for your question.