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A326373
Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.
3
1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299
OFFSET
0,4
COMMENTS
A set system (set of sets) is intersecting if no two edges are disjoint.
FORMULA
a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - Andrew Howroyd, Aug 12 2019
EXAMPLE
The a(3) = 3 intersecting set systems with empty intersection:
{}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
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], {1, n}], Intersection[#1, #2]=={}&], And[#=={}||Intersection@@#=={}]&]], {n, 0, 4}]
CROSSREFS
The inverse binomial transform is the covering case A326364.
Set systems with empty intersection are A318129.
Intersecting set systems are A051185.
Intersecting antichains with empty intersection are A326366.
Sequence in context: A269553 A361886 A086207 * A092052 A139999 A140870
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 01 2019
EXTENSIONS
a(6)-a(9) from Andrew Howroyd, Aug 12 2019
STATUS
approved