cp's OEIS Frontend

A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

%I A323299 #14 Aug 15 2019 14:37:02
%S A323299 1,1,1,2,5,26,261,3216,19617,80860,262651,737716,1920821,5013152,
%T A323299 14277485,47610876,186355041,820625616,3869589607,19039193980,
%U A323299 96332399701,499138921736,2639262062801,14234781051932,78188865206145,437305612997376,2487692697142251
%N A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.
%F A323299 Binomial transform of A323298.
%e A323299 The a(5) = 26 hypergraphs:
%e A323299   {}
%e A323299   {{1,2,3}}
%e A323299   {{1,2,4}}
%e A323299   {{1,2,5}}
%e A323299   {{1,3,4}}
%e A323299   {{1,3,5}}
%e A323299   {{1,4,5}}
%e A323299   {{2,3,4}}
%e A323299   {{2,3,5}}
%e A323299   {{2,4,5}}
%e A323299   {{3,4,5}}
%e A323299   {{1,2,3},{1,4,5}}
%e A323299   {{1,2,3},{2,4,5}}
%e A323299   {{1,2,3},{3,4,5}}
%e A323299   {{1,2,4},{1,3,5}}
%e A323299   {{1,2,4},{2,3,5}}
%e A323299   {{1,2,4},{3,4,5}}
%e A323299   {{1,2,5},{1,3,4}}
%e A323299   {{1,2,5},{2,3,4}}
%e A323299   {{1,2,5},{3,4,5}}
%e A323299   {{1,3,4},{2,3,5}}
%e A323299   {{1,3,4},{2,4,5}}
%e A323299   {{1,3,5},{2,3,4}}
%e A323299   {{1,3,5},{2,4,5}}
%e A323299   {{1,4,5},{2,3,4}}
%e A323299   {{1,4,5},{2,3,5}}
%e A323299 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.
%e A323299     1 X {}
%e A323299    35 X {{1,2,3}}
%e A323299   315 X {{1,2,5},{3,4,5}}
%e A323299   105 X {{1,2,7},{3,4,7},{5,6,7}}
%e A323299   840 X {{1,3,5},{2,3,6},{4,5,6}}
%e A323299   840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323299   210 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
%e A323299   630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323299   210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%e A323299    30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
%t A323299 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 A323299 Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&]],{n,8}]
%Y A323299 Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323298.
%K A323299 nonn
%O A323299 0,4
%A A323299 _Gus Wiseman_, Jan 11 2019
%E A323299 Terms a(11) and beyond from _Andrew Howroyd_, Aug 14 2019