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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 (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..5.

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}]

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: A179420 A080487 A077696 * A117271 A289998 A180353

Adjacent sequences:  A326358 A326359 A326360 * A326362 A326363 A326364

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 01 2019

STATUS

approved

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Last modified February 21 20:42 EST 2020. Contains 332111 sequences. (Running on oeis4.)