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%I #17 Oct 12 2023 08:10:38
%S 1,1,1,2,5,26,271,5596,231577,21286940,4392750641,2100400533176
%N Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.
%e The a(5) = 26 hypergraphs:
%e {}
%e {{1,2,3}}
%e {{1,2,4}}
%e {{1,2,5}}
%e {{1,3,4}}
%e {{1,3,5}}
%e {{1,4,5}}
%e {{2,3,4}}
%e {{2,3,5}}
%e {{2,4,5}}
%e {{3,4,5}}
%e {{1,2,3},{1,4,5}}
%e {{1,2,3},{2,4,5}}
%e {{1,2,3},{3,4,5}}
%e {{1,2,4},{1,3,5}}
%e {{1,2,4},{2,3,5}}
%e {{1,2,4},{3,4,5}}
%e {{1,2,5},{1,3,4}}
%e {{1,2,5},{2,3,4}}
%e {{1,2,5},{3,4,5}}
%e {{1,3,4},{2,3,5}}
%e {{1,3,4},{2,4,5}}
%e {{1,3,5},{2,3,4}}
%e {{1,3,5},{2,4,5}}
%e {{1,4,5},{2,3,4}}
%e {{1,4,5},{2,3,5}}
%e Non-isomorphic representatives of the 6 unlabeled 3-uniform hypertrees spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 271:
%e 1 X {}
%e 20 X {{1,2,3}}
%e 90 X {{1,2,5},{3,4,5}}
%e 10 X {{1,2,3},{4,5,6}}
%e 120 X {{1,3,5},{2,3,6},{4,5,6}}
%e 30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
%t stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>1&]],{n,8}]
%Y Essentially the same as A287232.
%Y Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jan 10 2019
%E a(9) from _Andrew Howroyd_, Aug 14 2019
%E a(10) and a(11) (using A287232) from _Joerg Arndt_, Oct 12 2023
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