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A326360
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Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.
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9
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OFFSET
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0,4
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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(1) = 1 through a(4) = 13 maximal antichains:
{} {12} {123} {1234}
{12}{13}{23} {12}{134}{234}
{13}{124}{234}
{14}{123}{234}
{23}{124}{134}
{24}{123}{134}
{34}{123}{124}
{12}{13}{14}{234}
{12}{23}{24}{134}
{13}{23}{34}{124}
{14}{24}{34}{123}
{123}{124}{134}{234}
{12}{13}{14}{23}{24}{34}
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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}], SubsetQ]]], {n, 0, 4}]
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PROG
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(Python)
# see Ignatov links
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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, A058891, A261005, A305000, A305844, A326358, 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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