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A324764 Number of anti-transitive rooted identity trees with n nodes.

%I A324764 #11 Jun 20 2020 02:04:59
%S A324764 1,1,1,1,3,4,9,20,41,89,196,443,987,2246,5114,11757,27122,62898,
%T A324764 146392,342204,802429,1887882
%N A324764 Number of anti-transitive rooted identity trees with n nodes.
%C A324764 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 A324764 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 A324764   {{{{{{{o}}}}}}}
%C A324764   {{{{{o,{o}}}}}}
%C A324764   {{{{o,{{o}}}}}}
%C A324764   {{{o,{{{o}}}}}}
%C A324764   {{{o,{o,{o}}}}}
%C A324764   {{{{o},{{o}}}}}
%C A324764   {{o,{{{{o}}}}}}
%C A324764   {{o,{{o,{o}}}}}
%C A324764   {{o,{o,{{o}}}}}
%C A324764   {{{o},{{{o}}}}}
%C A324764   {{{o},{o,{o}}}}
%C A324764   {{o,{o},{{o}}}}
%C A324764   {o,{{{{{o}}}}}}
%C A324764   {o,{{{o,{o}}}}}
%C A324764   {o,{{o,{{o}}}}}
%C A324764   {o,{{o},{{o}}}}
%C A324764   {{o},{{{{o}}}}}
%C A324764   {{o},{{o,{o}}}}
%C A324764   {{o},{o,{{o}}}}
%C A324764   {{{o}},{o,{o}}}
%H A324764 Gus Wiseman, <a href="/A324764/a324764.png">The a(9) = 41 anti-transitive rooted identity trees</a>.
%e A324764 The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
%e A324764   o  (o)  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o(o(o))))
%e A324764                           (o((o)))   ((o((o))))   (o((o(o))))
%e A324764                           ((((o))))  (o(((o))))   ((((o(o)))))
%e A324764                                      (((((o)))))  (((o)((o))))
%e A324764                                                   (((o((o)))))
%e A324764                                                   ((o)(((o))))
%e A324764                                                   ((o(((o)))))
%e A324764                                                   (o((((o)))))
%e A324764                                                   ((((((o))))))
%t A324764 idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
%t A324764 Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]
%Y A324764 Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
%Y A324764 Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.
%K A324764 nonn,more
%O A324764 1,5
%A A324764 _Gus Wiseman_, Mar 17 2019
%E A324764 a(21)-a(22) from _Jinyuan Wang_, Jun 20 2020