OFFSET
0,3
COMMENTS
Same as A059523 except with a(1) = 1 instead of 2.
Alternatively, these are 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. An antichain is a set of sets, none of which is a subset of any other.
FORMULA
Inverse binomial transform of A326965.
EXAMPLE
The a(3) = 36 set-systems:
{{1}{2}{3}} {{12}{13}{23}{123}} {{2}{3}{12}{13}{23}}
{{12}{13}{23}} {{1}{2}{3}{12}{13}} {{2}{3}{12}{13}{123}}
{{1}{2}{3}{12}} {{1}{2}{3}{12}{23}} {{2}{12}{13}{23}{123}}
{{1}{2}{3}{13}} {{1}{2}{3}{13}{23}} {{3}{12}{13}{23}{123}}
{{1}{2}{3}{23}} {{1}{2}{12}{13}{23}} {{1}{2}{3}{12}{13}{23}}
{{1}{2}{13}{23}} {{1}{2}{3}{12}{123}} {{1}{2}{3}{12}{13}{123}}
{{1}{2}{3}{123}} {{1}{2}{3}{13}{123}} {{1}{2}{3}{12}{23}{123}}
{{1}{3}{12}{23}} {{1}{2}{3}{23}{123}} {{1}{2}{3}{13}{23}{123}}
{{2}{3}{12}{13}} {{1}{3}{12}{13}{23}} {{1}{2}{12}{13}{23}{123}}
{{1}{12}{13}{23}} {{1}{2}{13}{23}{123}} {{1}{3}{12}{13}{23}{123}}
{{2}{12}{13}{23}} {{1}{3}{12}{23}{123}} {{2}{3}{12}{13}{23}{123}}
{{3}{12}{13}{23}} {{1}{12}{13}{23}{123}} {{1}{2}{3}{12}{13}{23}{123}}
MATHEMATICA
tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]], Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&tmQ[#]&]], {n, 0, 3}]
CROSSREFS
Covering set-systems are A003465.
Covering T_0 set-systems are A059201.
The version with empty edges allowed is A326960.
The non-covering version is A326965.
Covering set-systems whose dual is a weak antichain are A326970.
The unlabeled version is A326974.
The BII-numbers of T_1 set-systems are A326979.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 12 2019
STATUS
approved