cp's OEIS Frontend

A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.

%I A323297 #14 Aug 18 2019 12:29:15
%S A323297 1,1,1,2,16,76,271,1212,10158,78290,503231,3495966,33016534,327625520,
%T A323297 3000119669,28185006956,308636238516,3631959615948,42031903439809,
%U A323297 493129893459310,6264992355842706,84639308481270656,1159506969481515271,16131054826385628592
%N A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.
%H A323297 Andrew Howroyd, <a href="/A323297/b323297.txt">Table of n, a(n) for n = 0..200</a>
%F A323297 Binomial transform of A323296.
%F A323297 E.g.f.: exp(x - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). - _Andrew Howroyd_, Aug 18 2019
%e A323297 The a(4) = 16 hypergraphs:
%e A323297   {}
%e A323297   {{1,2,3}}
%e A323297   {{1,2,4}}
%e A323297   {{1,3,4}}
%e A323297   {{2,3,4}}
%e A323297   {{1,2,3},{1,2,4}}
%e A323297   {{1,2,3},{1,3,4}}
%e A323297   {{1,2,3},{2,3,4}}
%e A323297   {{1,2,4},{1,3,4}}
%e A323297   {{1,2,4},{2,3,4}}
%e A323297   {{1,3,4},{2,3,4}}
%e A323297   {{1,2,3},{1,2,4},{1,3,4}}
%e A323297   {{1,2,3},{1,2,4},{2,3,4}}
%e A323297   {{1,2,3},{1,3,4},{2,3,4}}
%e A323297   {{1,2,4},{1,3,4},{2,3,4}}
%e A323297   {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%e A323297 The following are non-isomorphic representatives of the 8 unlabeled 3-uniform hypergraphs on 6 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 271:
%e A323297    1 X {}
%e A323297   20 X {{1,2,3}}
%e A323297   90 X {{1,3,4},{2,3,4}}
%e A323297   10 X {{1,2,3},{4,5,6}}
%e A323297   60 X {{1,4,5},{2,4,5},{3,4,5}}
%e A323297   60 X {{1,2,4},{1,3,4},{2,3,4}}
%e A323297   15 X {{1,5,6},{2,5,6},{3,5,6},{4,5,6}}
%e A323297   15 X {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%t A323297 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 A323297 Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&]],{n,8}]
%o A323297 (PARI) seq(n)={Vec(serlaplace(exp(x - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ _Andrew Howroyd_, Aug 18 2019
%Y A323297 Cf. A000665, A025035, A125791, A289837, A299471, A302374, A302394, A306021, A323293, A323294, A323296.
%K A323297 nonn
%O A323297 0,4
%A A323297 _Gus Wiseman_, Jan 11 2019
%E A323297 a(10)-a(11) from _Alois P. Heinz_, Aug 11 2019
%E A323297 Terms a(12) and beyond from _Andrew Howroyd_, Aug 18 2019