A327062 - OEIS

A327062

Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.

1, 2, 5, 16, 81, 2595

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OFFSET

0,2

COMMENTS

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

LINKS

Table of n, a(n) for n=0..5.

EXAMPLE

The a(0) = 1 through a(3) = 16 antichains:

{} {} {} {}

{{1},{2}} {{1,2}}

{{1},{2}}

{{1},{3}}

{{2},{3}}

{{1},{2,3}}

{{2},{1,3}}

{{3},{1,2}}

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

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

MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

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]]]];

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]

CROSSREFS

Antichains are A000372.

The covering case is A319639.

The non-isomorphic multiset partition version is A319721.

The BII-numbers of these set-systems are the intersection of A326910 and A326853.

Set-systems whose dual is a weak antichain are A326968.

The unlabeled version is A327018.

Cf. A006126, A318099, A293606, A326704, A326950, A327020, A327057.

Sequence in context: A290609 A048754 A176343 * A334157 A306020 A357341

Adjacent sequences: A327059 A327060 A327061 * A327063 A327064 A327065

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 18 2019

STATUS

approved