
COMMENTS

A setsystem is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a setsystem 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.


MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==wQ[r, w]Q[w, r]], Q]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], Intersection[#1, #2]=={}&], Union@@#==Range[n]&&stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]


CROSSREFS

Covering intersecting setsystems are A305843.
The BIInumbers of these setsystems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The noncovering version is A327059.
The unlabeled multiset partition version is A327060.
Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.
Sequence in context: A156990 A075514 A344898 * A087306 A278877 A203682
Adjacent sequences: A327055 A327056 A327057 * A327059 A327060 A327061
