A326372
Number of intersecting antichains of (possibly empty) subsets of {1..n}.
Original entry on oeis.org
2, 3, 5, 13, 82, 2647, 1422565, 229809982113, 423295099074735261881
Offset: 0
The a(0) = 2 through a(3) = 13 antichains:
{} {} {} {}
{{}} {{}} {{}} {{}}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1,2}} {{3}}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
The case without empty edges is
A001206.
The inverse binomial transform is the spanning case
A305844.
Maximal intersecting antichains are
A326363.
Intersecting set systems are
A051185.
A326373
Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.
Original entry on oeis.org
1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299
Offset: 0
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}}
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.
Cf.
A000371,
A006126,
A007363,
A014466,
A058891,
A305844,
A307249,
A318128,
A326361,
A326362,
A326363,
A326365.
-
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}]
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