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A323296
Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.
5
1, 0, 0, 1, 11, 10, 25, 406, 4823, 15436, 72915, 895180, 11320441, 71777498, 519354927, 6155284240, 82292879425, 788821735656, 7772567489083, 98329764933354, 1400924444610675, 17424772471470490, 216091776292721021, 3035845122991962688, 46700545575567202903
OFFSET
0,5
COMMENTS
The only way to meet the requirements is to cover the vertices with zero or more disconnected 3-uniform hypergraphs with each edge having exactly two vertices in common (A323294). - Andrew Howroyd, Aug 18 2019
LINKS
FORMULA
From Andrew Howroyd, Aug 18 2019: (Start)
Exponential transform of A323294.
E.g.f.: exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). (End)
EXAMPLE
The a(4) = 11:
{{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 3 unlabeled 3-uniform hypergraphs spanning 7 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 406.
210 X {{1,2,3},{4,6,7},{5,6,7}}
140 X {{1,2,3},{4,5,7},{4,6,7},{5,6,7}}
21 X {{1,6,7},{2,6,7},{3,6,7},{4,6,7},{5,6,7}}
35 X {{1,2,3},{4,5,6},{4,5,7},{4,6,7},{5,6,7}}
MAPLE
b:= n-> `if`(n<5, (n-2)*(2*n^2-6*n+3)/6, n/2)*(n-1):
a:= proc(n) option remember; `if`(n=0, 1, add(
binomial(n-1, k-1)*b(k)*a(n-k), k=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Aug 18 2019
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[Select[stableSets[Subsets[Range[n], {3}], Length[Intersection[#1, #2]]==1&], Union@@#==Range[n]&]], {n, 8}]
PROG
(PARI) seq(n)={Vec(serlaplace(exp(-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(11) from Alois P. Heinz, Aug 12 2019
Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019
STATUS
approved