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A326361
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Number of maximal intersecting antichains of sets covering n vertices with no singletons.
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13
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OFFSET
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0,4
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COMMENTS
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Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.
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LINKS
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EXAMPLE
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The a(4) = 12 antichains:
{{1,2,3,4}}
{{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[Select[stableSets[Subsets[Range[n]], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&], Union@@#==Range[n]&]]], {n, 0, 5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
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CROSSREFS
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Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326362, A326363.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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