OFFSET
0,3
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}}. This sequence counts covering set-systems whose dual is strict and pairwise intersecting.
FORMULA
Inverse binomial transform of A327052.
EXAMPLE
The a(1) = 1 through a(2) = 3 set-systems:
{} {{1}} {{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}
The a(3) = 62 set-systems:
1 2 123 1 2 3 123 1 2 12 13 23 1 2 3 12 13 23 1 2 3 12 13 23 123
1 3 123 1 12 13 23 1 2 3 12 123 1 2 3 12 13 123
2 3 123 1 2 12 123 1 2 3 13 123 1 2 3 12 23 123
1 12 123 1 2 13 123 1 2 3 23 123 1 2 3 13 23 123
1 13 123 1 2 23 123 1 3 12 13 23 1 2 12 13 23 123
12 13 23 1 3 12 123 2 3 12 13 23 1 3 12 13 23 123
2 12 123 1 3 13 123 1 2 12 13 123 2 3 12 13 23 123
2 23 123 1 3 23 123 1 2 12 23 123
3 13 123 2 12 13 23 1 2 13 23 123
3 23 123 2 3 12 123 1 3 12 13 123
12 13 123 2 3 13 123 1 3 12 23 123
12 23 123 2 3 23 123 1 3 13 23 123
13 23 123 3 12 13 23 2 3 12 13 123
1 12 13 123 2 3 12 23 123
1 12 23 123 2 3 13 23 123
1 13 23 123 1 12 13 23 123
2 12 13 123 2 12 13 23 123
2 12 23 123 3 12 13 23 123
2 13 23 123
3 12 13 123
3 12 23 123
3 13 23 123
12 13 23 123
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}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 3}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 18 2019
EXTENSIONS
a(5)-a(7) from Christian Sievers, Feb 04 2024
STATUS
approved