A326375 - OEIS

A326375

Number of intersecting antichains of subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

2, 2, 2, 3, 29, 1961, 1379274, 229755337550, 423295079757497714060

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OFFSET

0,1

COMMENTS

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

LINKS

Table of n, a(n) for n=0..8.

FORMULA

a(n) = A326366(n) + 1.

EXAMPLE

The a(4) = 29 antichains:

{}

{{}}

{{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},{2,3,4}}

{{1,2},{1,4},{2,3,4}}

{{1,2},{2,3},{1,3,4}}

{{1,2},{2,4},{1,3,4}}

{{1,3},{1,4},{2,3,4}}

{{1,3},{2,3},{1,2,4}}

{{1,3},{3,4},{1,2,4}}

{{1,4},{2,4},{1,2,3}}

{{1,4},{3,4},{1,2,3}}

{{2,3},{2,4},{1,3,4}}

{{2,3},{3,4},{1,2,4}}

{{2,4},{3,4},{1,2,3}}

{{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]]]];

Table[Length[Select[stableSets[Subsets[Range[n]], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&], #=={}||Intersection@@#=={}&]], {n, 0, 4}]

CROSSREFS

The case without empty edges is A326366.

Intersecting antichains are A326372.

Antichains of nonempty sets with empty intersection are A006126 or A307249.

Cf. A001206, A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326363, A326365, A326373.

Sequence in context: A051007 A240166 A346801 * A358318 A071470 A197116

Adjacent sequences: A326372 A326373 A326374 * A326376 A326377 A326378

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 03 2019

EXTENSIONS

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

STATUS

approved