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COMMENTS
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A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence counts pairwise intersecting set-systems that are cointersecting, meaning their dual is pairwise intersecting.
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EXAMPLE
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The a(0) = 1 through a(2) = 6 set-systems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
The a(3) = 34 set-systems:
{} {{1}} {{1}{12}} {{1}{12}{123}} {{1}{12}{13}{123}}
{{2}} {{1}{13}} {{1}{13}{123}} {{2}{12}{23}{123}}
{{3}} {{2}{12}} {{12}{13}{23}} {{3}{13}{23}{123}}
{{12}} {{2}{23}} {{2}{12}{123}} {{12}{13}{23}{123}}
{{13}} {{3}{13}} {{2}{23}{123}}
{{23}} {{3}{23}} {{3}{13}{123}}
{{123}} {{1}{123}} {{3}{23}{123}}
{{2}{123}} {{12}{13}{123}}
{{3}{123}} {{12}{23}{123}}
{{12}{123}} {{13}{23}{123}}
{{13}{123}}
{{23}{123}}
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MATHEMATICA
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dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
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]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], Intersection[#1, #2]=={}&], stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 4}]
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