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A323297
Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.
4
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
0,4
LINKS
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
EXAMPLE
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}}
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}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 11 2019
EXTENSIONS
a(10)-a(11) from Alois P. Heinz, Aug 11 2019
Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019
STATUS
approved