OFFSET
1,11
COMMENTS
A finitary set is transitive if every element is also a subset. Transitive sets are also called full sets.
LINKS
Wikipedia, Transitive set
Gus Wiseman, Transitive rooted identity trees example n=23
EXAMPLE
Sequence of transitive finitary sets begins:
1 ()
2 (())
4 (()(()))
7 (()(())((())))
8 (()(())(()(())))
11 (()(())((()))(((()))))
(()(())((()))(()(())))
12 (()(())((()))(()((()))))
13 (()(())((()))((())((()))))
(()(())(()(()))((()(()))))
14 (()(())((()))(()(())((()))))
(()(())(()(()))(()(()(()))))
15 (()(())((()))(((())))(()(())))
(()(())(()(()))((())(()(()))))
16 (()(())((()))(((())))((((())))))
(()(())((()))(((())))(()((()))))
(()(())((()))(()(()))(()((()))))
(()(())((()))(()(()))((()(()))))
(()(())(()(()))(()(())(()(()))))
17 (()(())((()))(((())))(()(((())))))
(()(())((()))(((())))((())((()))))
(()(())((()))(()(()))(()(()(()))))
(()(())((()))(()(()))((())((()))))
18 (()(())((()))(((())))((())(((())))))
(()(())((()))(((())))(()(())((()))))
(()(())((()))(()(()))((())(()(()))))
(()(())((()))(()(()))(()(())((()))))
(()(())((()))((()((()))))(()((()))))
(()(())((()))(()((())))((())((()))))
MATHEMATICA
transfins[n_]:=transfins[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[nov, _List, {0, Infinity}]<=n]]]/@#, #]&, Array[transfins, n-1, 1, Union]], Count[#, _List, {0, Infinity}]===n&]];
Table[Length[transfins[n]], {n, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 21 2016
STATUS
approved