A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.
1, 1, 1, 2, 16, 76, 271, 1212, 10158, 78290, 503231, 3495966, 33016534, 327625520, 3000119669, 28185006956, 308636238516, 3631959615948, 42031903439809, 493129893459310, 6264992355842706, 84639308481270656, 1159506969481515271, 16131054826385628592
Offset: 0
Keywords
Examples
The a(4) = 16 hypergraphs:
{}
{{1,2,3}}
{{1,2,4}}
{{1,3,4}}
{{2,3,4}}
{{1,2,3},{1,2,4}}
{{1,2,3},{1,3,4}}
{{1,2,3},{2,3,4}}
{{1,2,4},{1,3,4}}
{{1,2,4},{2,3,4}}
{{1,3,4},{2,3,4}}
{{1,2,3},{1,2,4},{1,3,4}}
{{1,2,3},{1,2,4},{2,3,4}}
{{1,2,3},{1,3,4},{2,3,4}}
{{1,2,4},{1,3,4},{2,3,4}}
{{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
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:
1 X {}
20 X {{1,2,3}}
90 X {{1,3,4},{2,3,4}}
10 X {{1,2,3},{4,5,6}}
60 X {{1,4,5},{2,4,5},{3,4,5}}
60 X {{1,2,4},{1,3,4},{2,3,4}}
15 X {{1,5,6},{2,5,6},{3,5,6},{4,5,6}}
15 X {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
Crossrefs
Programs
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Mathematica
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]]]]; Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&]],{n,8}] -
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
Formula
Binomial transform of A323296.
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
Extensions
a(10)-a(11) from Alois P. Heinz, Aug 11 2019
Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019