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A327062
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Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.
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9
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OFFSET
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0,2
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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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(3) = 16 antichains:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1,2}} {{3}}
{{1},{2}} {{1,2}}
{{1,3}}
{{2,3}}
{{1},{2}}
{{1,2,3}}
{{1},{3}}
{{2},{3}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{2},{3}}
{{1,2},{1,3},{2,3}}
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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}], SubsetQ], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]
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CROSSREFS
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The non-isomorphic multiset partition version is A319721.
The BII-numbers of these set-systems are the intersection of A326910 and A326853.
Set-systems whose dual is a weak antichain are A326968.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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