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A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

%I A323292 #12 Aug 15 2019 01:39:26
%S A323292 1,0,0,1,0,15,160,4125,193200,19384225
%N A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.
%F A323292 Inverse binomial transform of A323293. - _Andrew Howroyd_, Aug 14 2019
%e A323292 The a(5) = 15 hypergraphs:
%e A323292   {{1,2,3},{1,4,5}}
%e A323292   {{1,2,3},{2,4,5}}
%e A323292   {{1,2,3},{3,4,5}}
%e A323292   {{1,2,4},{1,3,5}}
%e A323292   {{1,2,4},{2,3,5}}
%e A323292   {{1,2,4},{3,4,5}}
%e A323292   {{1,2,5},{1,3,4}}
%e A323292   {{1,2,5},{2,3,4}}
%e A323292   {{1,2,5},{3,4,5}}
%e A323292   {{1,3,4},{2,3,5}}
%e A323292   {{1,3,4},{2,4,5}}
%e A323292   {{1,3,5},{2,3,4}}
%e A323292   {{1,3,5},{2,4,5}}
%e A323292   {{1,4,5},{2,3,4}}
%e A323292   {{1,4,5},{2,3,5}}
%e A323292 Non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs 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) = 160:
%e A323292    10 X {{1,2,3},{4,5,6}}
%e A323292   120 X {{1,3,5},{2,3,6},{4,5,6}}
%e A323292    30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
%t A323292 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 A323292 Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>=2&],Union@@#==Range[n]&]],{n,6}]
%Y A323292 Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A320395, A322451, A323293-A323299.
%K A323292 nonn,more
%O A323292 0,6
%A A323292 _Gus Wiseman_, Jan 10 2019
%E A323292 a(9) from _Andrew Howroyd_, Aug 14 2019