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%I #5 Dec 21 2016 11:06:13
%S 0,1,0,1,0,0,0,1,0,0,1,0,0,1,0,1,0,2,2,1,1,4,3,4,2,5,6,10,8,11,11,20,
%T 22,29,36,45,53,77,83,108,141,172,208,274,323
%N Number of maximal transitive finitary sets with n brackets.
%C A finitary set is transitive if every element is also a subset. A set system is maximal if the union is also a member.
%H Gus Wiseman, <a href="/A279863/a279863.png">Maximal transitive identity trees up to n=25</a>
%e The a(23)=3 maximal transitive finitary sets are:
%e (()(())(()(()))((())(()(())))(()(())(()(())))),
%e (()(())((()))(((())))(()((())))(()(())((())))),
%e (()(())((()))(()(()))(()((())))(()(())((())))).
%t maxtransfins[n_]:=If[n===1,{},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[Union[nov,{Union@@nov}],_List,{0,Infinity}]<=n]]]/@#,#]&,{{}}],And[Count[#,_List,{0,Infinity}]===n,MemberQ[#,Union@@#]]&]];
%t Table[Length[maxtransfins[n]],{n,20}]
%Y Cf. A001192, A004111, A276625, A279065, A279614, A279861.
%K nonn
%O 1,18
%A _Gus Wiseman_, Dec 21 2016
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