OFFSET
1,3
COMMENTS
We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
LINKS
EXAMPLE
The standard ordered rooted tree ranking begins:
1: o 10: (((o))o) 19: (((o))(o))
2: (o) 11: ((o)(o)) 20: (((o))oo)
3: ((o)) 12: ((o)oo) 21: ((o)((o)))
4: (oo) 13: (o((o))) 22: ((o)(o)o)
5: (((o))) 14: (o(o)o) 23: ((o)o(o))
6: ((o)o) 15: (oo(o)) 24: ((o)ooo)
7: (o(o)) 16: (oooo) 25: (o(oo))
8: (ooo) 17: ((((o)))) 26: (o((o))o)
9: ((oo)) 18: ((oo)o) 27: (o(o)(o))
For example, the 52nd ordered tree is (o((o))oo) so a(52) = 3.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Table[Depth[srt[n]]-2, {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 16 2022
STATUS
approved