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A305001 Number of labeled antichains of finite sets spanning n vertices without singletons. 23
1, 0, 1, 5, 87, 6398, 7745253, 2414573042063, 56130437190053518791691, 286386577668298410118121281898931424413687 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
From Gus Wiseman, Jul 03 2019: (Start)
Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}
{{1,2},{1,3},{2,3}}
(End)
LINKS
EXAMPLE
The a(3) = 5 antichains:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
MATHEMATICA
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]]]];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], And[Union@@#==Range[n], #=={}||Intersection@@#=={}]&]], {n, 0, 5}] (* Gus Wiseman, Jul 03 2019 *)
CROSSREFS
The binomial transform is the non-covering case A307249.
The second binomial transform is A014466.
Sequence in context: A069948 A316727 A216088 * A324092 A208019 A297529
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 23 2018
EXTENSIONS
a(9) from A307249 - Dmitry I. Ignatov, Nov 27 2023
STATUS
approved

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Last modified February 27 07:37 EST 2024. Contains 370367 sequences. (Running on oeis4.)