OFFSET
0,3
COMMENTS
A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges 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.
FORMULA
Inverse binomial transform of A326968.
EXAMPLE
The a(3) = 43 set-systems:
{123} {1}{23} {1}{2}{3} {1}{2}{3}{12}
{2}{13} {12}{13}{23} {1}{2}{3}{13}
{3}{12} {1}{23}{123} {1}{2}{3}{23}
{2}{13}{123} {1}{2}{13}{23}
{3}{12}{123} {1}{2}{3}{123}
{1}{3}{12}{23}
{2}{3}{12}{13}
{1}{12}{13}{23}
{2}{12}{13}{23}
{3}{12}{13}{23}
{12}{13}{23}{123}
.
{1}{2}{3}{12}{13} {1}{2}{3}{12}{13}{23} {1}{2}{3}{12}{13}{23}{123}
{1}{2}{3}{12}{23} {1}{2}{3}{12}{13}{123}
{1}{2}{3}{13}{23} {1}{2}{3}{12}{23}{123}
{1}{2}{12}{13}{23} {1}{2}{3}{13}{23}{123}
{1}{2}{3}{12}{123} {1}{2}{12}{13}{23}{123}
{1}{2}{3}{13}{123} {1}{3}{12}{13}{23}{123}
{1}{2}{3}{23}{123} {2}{3}{12}{13}{23}{123}
{1}{3}{12}{13}{23}
{2}{3}{12}{13}{23}
{1}{2}{13}{23}{123}
{1}{3}{12}{23}{123}
{2}{3}{12}{13}{123}
{1}{12}{13}{23}{123}
{2}{12}{13}{23}{123}
{3}{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]&&stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]
CROSSREFS
Covering set-systems are A003465.
Covering set-systems whose dual is strict are A059201.
The T_1 case is A326961.
The BII-numbers of these set-systems are A326966.
The non-covering case is A326968.
The unlabeled version is A326973.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 10 2019
STATUS
approved