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EXAMPLE
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The a(5) = 26 hypergraphs:
{}
{{1,2,3}}
{{1,2,4}}
{{1,2,5}}
{{1,3,4}}
{{1,3,5}}
{{1,4,5}}
{{2,3,4}}
{{2,3,5}}
{{2,4,5}}
{{3,4,5}}
{{1,2,3},{1,4,5}}
{{1,2,3},{2,4,5}}
{{1,2,3},{3,4,5}}
{{1,2,4},{1,3,5}}
{{1,2,4},{2,3,5}}
{{1,2,4},{3,4,5}}
{{1,2,5},{1,3,4}}
{{1,2,5},{2,3,4}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,5}}
{{1,3,4},{2,4,5}}
{{1,3,5},{2,3,4}}
{{1,3,5},{2,4,5}}
{{1,4,5},{2,3,4}}
{{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 10 unlabeled 3-uniform hypergraphs on 7 vertices where every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 3216.
1 X {}
35 X {{1,2,3}}
315 X {{1,2,5},{3,4,5}}
105 X {{1,2,7},{3,4,7},{5,6,7}}
840 X {{1,3,5},{2,3,6},{4,5,6}}
840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
210 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
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