|
|
A305001
|
|
Number of labeled antichains of finite sets spanning n vertices without singletons.
|
|
23
|
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
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.
Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|