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A326974
Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.
17
1, 1, 2, 16, 1212
OFFSET
0,3
COMMENTS
Alternatively, these are unlabeled set-systems covering n vertices whose dual is a (strict) antichain. 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}}. An antichain is a set-system where no edge is a subset of any other.
FORMULA
a(n > 0) = A326972(n) - A326972(n - 1).
EXAMPLE
Non-isomorphic representatives of the a(0) = 1 through a(3) = 16 set-systems:
{} {{1}} {{1},{2}} {{1},{2},{3}}
{{1},{2},{1,2}} {{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{1,3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{2,3},{1,2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,3},{2,3},{1,2,3}}
{{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
CROSSREFS
Unlabeled covers are A055621.
The same with T_0 instead of T_1 is A319637.
The labeled version is A326961.
The non-covering version is A326972 (partial sums).
Unlabeled covering set-systems whose dual is a weak antichain are A326973.
Sequence in context: A125791 A102103 A337070 * A060597 A091479 A016031
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 11 2019
STATUS
approved