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A007363
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Maximal self-dual antichains on n points.
(Formerly M2505)
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15
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OFFSET
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1,3
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COMMENTS
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If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:
{1} {1} {1} {1}
{2} {2} {2}
{12} {3} {3}
{123} {4}
{12}{13}{23} {1234}
{12}{13}{23}
{12}{14}{24}
{13}{14}{34}
{23}{24}{34}
{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}
(End)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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], {1, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]]], {n, 0, 5}] (* Gus Wiseman, Jul 02 2019 *)
(* 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 = FindClique[g, Infinity, All];
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CROSSREFS
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Intersecting antichains are A326372.
Intersecting antichains of nonempty sets are A001206.
Unlabeled intersecting antichains are A305857.
Maximal antichains of nonempty sets are A326359.
The case with empty edges allowed is 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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