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A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain. 4
1, 1, 1, 3, 155 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
LINKS
FORMULA
Inverse binomial transform of A327059.
EXAMPLE
The a(0) = 1 through a(3) = 3 set-systems:
{} {{1}} {{12}} {{123}}
{{12}{13}{23}}
{{12}{13}{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==w||Q[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 set-systems are A305843.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The non-covering version is A327059.
The unlabeled multiset partition version is A327060.
Sequence in context: A156990 A075514 A344898 * A087306 A278877 A203682
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 18 2019
STATUS
approved

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Last modified April 15 11:23 EDT 2024. Contains 371681 sequences. (Running on oeis4.)