A326968 Number of set-systems on n vertices whose dual is a weak antichain.

1, 2, 6, 56, 19446

Offset

0,2

Comments

A set-system is a finite set of finite nonempty sets. 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..4.

Formula

a(n) = A326969(n)/2.

Binomial transform of A326970.

Example

The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{2},{1,2}}

Mathematica

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]

Crossrefs

The case with strict dual is A326965.

The BII-numbers of these set-systems are A326966.

The version with empty edges allowed is A326969.

The covering case is A326970.

The unlabeled version is A326971.

Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.

Sequence in context: A365776 A198445 A248377 * A209497 A209738 A209459

Adjacent sequences: A326965 A326966 A326967 * A326969 A326970 A326971

Keyword

nonn,more

Author

Gus Wiseman, Aug 10 2019

Status

approved