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A279861
Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.
26
1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 2, 2, 5, 4, 6, 8, 10, 14, 23, 26, 34, 46, 64, 81, 115, 158, 199, 277, 376, 505, 684, 934, 1241, 1711, 2300, 3123, 4236, 5763, 7814, 10647, 14456, 19662
OFFSET
1,11
COMMENTS
A finitary set is transitive if every element is also a subset. Transitive sets are also called full sets.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 21 2016
STATUS
approved