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A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

%I A323298 #17 Feb 12 2022 20:29:03
%S A323298 1,0,0,1,0,15,150,1815,0,945,0,10395,0,135135,0,2027025,0,34459425,0,
%T A323298 654729075,0,13749310575,0,316234143225,0,7905853580625,0,
%U A323298 213458046676875,0,6190283353629375,0,191898783962510625,0,6332659870762850625,0,221643095476699771875
%N A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.
%C A323298 The only way to cover more than 7 vertices is with edges all having a single common vertex. For the special cases of n = 6 or n = 7, there are also covers without a common vertex. - _Andrew Howroyd_, Aug 15 2019
%H A323298 Andrew Howroyd, <a href="/A323298/b323298.txt">Table of n, a(n) for n = 0..200</a>
%F A323298 a(2*n) = 0 for n > 3; a(2*n-1) = A001147(n) for n > 4. - _Andrew Howroyd_, Aug 15 2019
%e A323298 The a(5) = 15 hypergraphs:
%e A323298   {{1,4,5},{2,3,5}}
%e A323298   {{1,4,5},{2,3,4}}
%e A323298   {{1,3,5},{2,4,5}}
%e A323298   {{1,3,5},{2,3,4}}
%e A323298   {{1,3,4},{2,4,5}}
%e A323298   {{1,3,4},{2,3,5}}
%e A323298   {{1,2,5},{3,4,5}}
%e A323298   {{1,2,5},{2,3,4}}
%e A323298   {{1,2,5},{1,3,4}}
%e A323298   {{1,2,4},{3,4,5}}
%e A323298   {{1,2,4},{2,3,5}}
%e A323298   {{1,2,4},{1,3,5}}
%e A323298   {{1,2,3},{3,4,5}}
%e A323298   {{1,2,3},{2,4,5}}
%e A323298   {{1,2,3},{1,4,5}}
%e A323298 The following are non-isomorphic representatives of the 5 unlabeled 3-uniform hypergraphs spanning 7 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 1815.
%e A323298   105 X {{1,2,7},{3,4,7},{5,6,7}}
%e A323298   840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323298   630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323298   210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323298    30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323298 From _Andrew Howroyd_, Aug 15 2019: (Start)
%e A323298 The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypergraphs spanning 6 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 150.
%e A323298     120 X {{1,2,3},{1,4,5},{3,5,6}}
%e A323298      30 X {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}
%e A323298 (End)
%t A323298 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 A323298 Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&],Union@@#==Range[n]&]],{n,10}]
%o A323298 (PARI) a(n)={if(n%2, if(n<=3, n==3, if(n==7, 1815, n!/(2^(n\2)*(n\2)!))), if(n==6, 150, n==0))} \\ _Andrew Howroyd_, Aug 15 2019
%Y A323298 Cf. A001147, A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323299.
%K A323298 nonn
%O A323298 0,6
%A A323298 _Gus Wiseman_, Jan 11 2019
%E A323298 Terms a(13) and beyond from _Andrew Howroyd_, Aug 15 2019