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EXAMPLE
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The a(4) = 16 maximal intersecting antichains:
{{1,2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,4},{2,4}}
{{1,3},{1,4},{3,4}}
{{2,3},{2,4},{3,4}}
{{1,2},{1,3,4},{2,3,4}}
{{1,3},{1,2,4},{2,3,4}}
{{1,4},{1,2,3},{2,3,4}}
{{2,3},{1,2,4},{1,3,4}}
{{2,4},{1,2,3},{1,3,4}}
{{3,4},{1,2,3},{1,2,4}}
{{1,2},{1,3},{1,4},{2,3,4}}
{{1,2},{2,3},{2,4},{1,3,4}}
{{1,3},{2,3},{3,4},{1,2,4}}
{{1,4},{2,4},{3,4},{1,2,3}}
{{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
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MATHEMATICA
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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]]]];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n], {2, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]]], {n, 0, 5}]
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