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A326362
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Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.
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12
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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 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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Table of n, a(n) for n=0..7.
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FORMULA
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For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.
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EXAMPLE
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The a(4) = 16 maximal intersecting antichains:
{{1,2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,4},{2,4}}
{{1,3},{1,4},{3,4}}
{{2,3},{2,4},{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[stableSets[Subsets[Range[n], {2, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]]], {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 = FindClique[g, Infinity, All];
Length[sets]-Log[2, n]-1 (* Elijah Beregovsky, May 06 2020 *)
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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, A326361, A326363.
Sequence in context: A309440 A226012 A011552 * A287222 A216598 A219397
Adjacent sequences: A326359 A326360 A326361 * A326363 A326364 A326365
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KEYWORD
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nonn,more
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AUTHOR
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Gus Wiseman, Jul 01 2019
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EXTENSIONS
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a(7) from Elijah Beregovsky, May 06 2020
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STATUS
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approved
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