OFFSET
1,5
COMMENTS
A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (o = {}):
{{{{{{{o}}}}}}}
{{{{{o,{o}}}}}}
{{{{o,{{o}}}}}}
{{{o,{{{o}}}}}}
{{{o,{o,{o}}}}}
{{{{o},{{o}}}}}
{{o,{{{{o}}}}}}
{{o,{{o,{o}}}}}
{{o,{o,{{o}}}}}
{{{o},{{{o}}}}}
{{{o},{o,{o}}}}
{{o,{o},{{o}}}}
{o,{{{{{o}}}}}}
{o,{{{o,{o}}}}}
{o,{{o,{{o}}}}}
{o,{{o},{{o}}}}
{{o},{{{{o}}}}}
{{o},{{o,{o}}}}
{{o},{o,{{o}}}}
{{{o}},{o,{o}}}
LINKS
EXAMPLE
The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
o (o) ((o)) (((o))) ((o(o))) (((o(o)))) ((o(o(o))))
(o((o))) ((o((o)))) (o((o(o))))
((((o)))) (o(((o)))) ((((o(o)))))
(((((o))))) (((o)((o))))
(((o((o)))))
((o)(((o))))
((o(((o)))))
(o((((o)))))
((((((o))))))
MATHEMATICA
idall[n_]:=If[n==1, {{}}, Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])], UnsameQ@@#&]];
Table[Length[Select[idall[n], Intersection[Union@@#, #]=={}&]], {n, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 17 2019
EXTENSIONS
a(21)-a(22) from Jinyuan Wang, Jun 20 2020
STATUS
approved