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 A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons. 13
 1, 1, 1, 2, 12, 133, 11386, 12143511 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Table of n, a(n) for n=0..7. EXAMPLE 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}} MATHEMATICA 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 &]; Length[sets] (* Elijah Beregovsky, May 05 2020 *) CROSSREFS 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. Sequence in context: A346186 A080487 A077696 * A117271 A289998 A180353 Adjacent sequences: A326358 A326359 A326360 * A326362 A326363 A326364 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 01 2019 EXTENSIONS a(6)-a(7) from Elijah Beregovsky, May 05 2020 STATUS approved

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Last modified September 22 10:30 EDT 2023. Contains 365520 sequences. (Running on oeis4.)