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A326358
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Number of maximal antichains of subsets of {1..n}.
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18
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OFFSET
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0,2
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COMMENTS
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A set system (set of sets) is an antichain if no element is a subset of any other.
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 1 through a(3) = 7 maximal antichains:
{} {} {} {}
{1} {12} {123}
{1}{2} {1}{23}
{2}{13}
{3}{12}
{1}{2}{3}
{12}{13}{23}
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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]], SubsetQ]]], {n, 0, 5}]
(* alternatively *)
maxachP[n_]:=FindIndependentVertexSet[
Flatten[Map[Function[s, Map[# \[DirectedEdge] s &, Most[Subsets[s]]]],
Subsets[Range[n]]]], Infinity, All];
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PROG
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(GAP) LoadPackage("grape");
maxachP:=function(n) local g, G;
g:=Graph(Group(()), Combinations([1..n]), function(x, g) return x; end,
function(x, y) return not IsSubset(x, y) and not IsSubset(y, x); end, true);
G:=AutGroupGraph(g);
return Sum(CompleteSubgraphs(NewGroupGraph(G, g), -1, 2),
function(c) return Length(Orbit(G, c, OnSets)); end);
end;
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CROSSREFS
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Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305001, A305844, A326360, A326361, 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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