
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}}. This sequence counts pairwise intersecting setsystems that are cointersecting, meaning their dual is pairwise intersecting.


EXAMPLE

The a(0) = 1 through a(2) = 6 setsystems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
The a(3) = 34 setsystems:
{} {{1}} {{1}{12}} {{1}{12}{123}} {{1}{12}{13}{123}}
{{2}} {{1}{13}} {{1}{13}{123}} {{2}{12}{23}{123}}
{{3}} {{2}{12}} {{12}{13}{23}} {{3}{13}{23}{123}}
{{12}} {{2}{23}} {{2}{12}{123}} {{12}{13}{23}{123}}
{{13}} {{3}{13}} {{2}{23}{123}}
{{23}} {{3}{23}} {{3}{13}{123}}
{{123}} {{1}{123}} {{3}{23}{123}}
{{2}{123}} {{12}{13}{123}}
{{3}{123}} {{12}{23}{123}}
{{12}{123}} {{13}{23}{123}}
{{13}{123}}
{{23}{123}}


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]=={}&], stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 4}]


CROSSREFS

Intersecting setsystems are A051185.
The unlabeled multiset partition version is A319765.
The BIInumbers of these setsystems are A326912.
The covering case is A327037.
Cointersecting setsystems are A327039.
The case where the dual is strict is A327040.
Cf. A058891, A319767, A319774, A326854, A327052.
Sequence in context: A075272 A353536 A224913 * A228931 A101262 A135965
Adjacent sequences: A327035 A327036 A327037 * A327039 A327040 A327041
