A324764 - OEIS

%I #11 Jun 20 2020 02:04:59

%S 1,1,1,1,3,4,9,20,41,89,196,443,987,2246,5114,11757,27122,62898,

%T 146392,342204,802429,1887882

%N Number of anti-transitive rooted identity trees with n nodes.

%C 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.

%C 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 = {}):

%C   {{{{{{{o}}}}}}}

%C   {{{{{o,{o}}}}}}

%C   {{{{o,{{o}}}}}}

%C   {{{o,{{{o}}}}}}

%C   {{{o,{o,{o}}}}}

%C   {{{{o},{{o}}}}}

%C   {{o,{{{{o}}}}}}

%C   {{o,{{o,{o}}}}}

%C   {{o,{o,{{o}}}}}

%C   {{{o},{{{o}}}}}

%C   {{{o},{o,{o}}}}

%C   {{o,{o},{{o}}}}

%C   {o,{{{{{o}}}}}}

%C   {o,{{{o,{o}}}}}

%C   {o,{{o,{{o}}}}}

%C   {o,{{o},{{o}}}}

%C   {{o},{{{{o}}}}}

%C   {{o},{{o,{o}}}}

%C   {{o},{o,{{o}}}}

%C   {{{o}},{o,{o}}}

%H Gus Wiseman, <a href="/A324764/a324764.png">The a(9) = 41 anti-transitive rooted identity trees</a>.

%e The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:

%e   o  (o)  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o(o(o))))

%e                           (o((o)))   ((o((o))))   (o((o(o))))

%e                           ((((o))))  (o(((o))))   ((((o(o)))))

%e                                      (((((o)))))  (((o)((o))))

%e                                                   (((o((o)))))

%e                                                   ((o)(((o))))

%e                                                   ((o(((o)))))

%e                                                   (o((((o)))))

%e                                                   ((((((o))))))

%t idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];

%t Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

%Y Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.

%Y Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

%K nonn,more

%O 1,5

%A _Gus Wiseman_, Mar 17 2019

%E a(21)-a(22) from _Jinyuan Wang_, Jun 20 2020