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A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain. 2
1, 2, 4, 10, 178 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
Binomial transform of A327058.
EXAMPLE
The a(0) = 1 through a(3) = 10 set-systems:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{12}} {{3}}
{{12}}
{{13}}
{{23}}
{{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]=={}&], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]
CROSSREFS
Intersecting set-systems are A051185.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Set-systems whose dual is a weak antichain are A326968.
The covering version is A327058.
The unlabeled multiset partition version is A327060.
Sequence in context: A223851 A297364 A355203 * A012555 A012722 A012616
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 18 2019
STATUS
approved

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Last modified March 19 07:49 EDT 2024. Contains 370958 sequences. (Running on oeis4.)