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A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}. 12
1, 1, 1, 2, 16, 163, 11742 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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..6.

FORMULA

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

EXAMPLE

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}}

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[stableSets[Subsets[Range[n], {2, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]]], {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, A326361, A326363.

Sequence in context: A309440 A226012 A011552 * A287222 A216598 A219397

Adjacent sequences:  A326359 A326360 A326361 * A326363 A326364 A326365

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 01 2019

STATUS

approved

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Last modified February 17 23:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)