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A326967
Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
5
2, 4, 10, 92, 38362, 4020654364, 18438434849260080818, 340282363593610212050791236025945013956, 115792089237316195072053288318104625957065868613454666314675263144628100544274
OFFSET
0,1
COMMENTS
Alternatively, these are sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. The dual of a set of subsets 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}}. An antichain is a set of sets, none of which is a subset of any other.
FORMULA
a(n) = 2 * A326965(n).
Binomial transform of A326960.
EXAMPLE
The a(0) = 2 through a(2) = 10 sets of subsets:
{} {} {}
{{}} {{}} {{}}
{{1}} {{1}}
{{},{1}} {{2}}
{{},{1}}
{{},{2}}
{{1},{2}}
{{},{1},{2}}
{{1},{2},{1,2}}
{{},{1},{2},{1,2}}
MATHEMATICA
tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]], Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n]]], tmQ[#]&]], {n, 0, 3}]
CROSSREFS
Sets of subsets are A001146.
The unlabeled version is A326951.
The covering version is A326960.
The case without empty edges is A326965.
Sets of subsets whose dual is a weak antichain are A326969.
Sequence in context: A090256 A270479 A126140 * A223851 A371621 A297364
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 10 2019
STATUS
approved