A358505 Binary encoding of the n-th standard ordered rooted tree.

0, 2, 12, 10, 56, 50, 44, 42, 52, 226, 204, 202, 184, 178, 172, 170, 240, 210, 908, 906, 824, 818, 812, 810, 180, 738, 716, 714, 696, 690, 684, 682, 228, 962, 844, 842, 3640, 3634, 3628, 3626, 820, 3298, 3276, 3274, 3256, 3250, 3244, 3242, 752, 722, 2956, 2954

Offset

1,2

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

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

Table of n, a(n) for n=1..52.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Example

The sixth standard tree is {{{}},{}}, which becomes (1,1,0,0,1,0), so a(6) = 50.

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
trt[t_]:=FromDigits[Take[DeleteCases[Characters[ToString[t]]/.{"{"->1, "}"->0}, ", "|" "], {2, -2}], 2];
Table[trt[srt[n]], {n, 100}]

Crossrefs

Sorting gives A014486.

A dual sequence is A358523.

Cf. A000108, A001263, A057122, A358371, A358372, A358373, A358379, A358453.

Sequence in context: A280015 A343645 A245281 * A308215 A216478 A181060

Adjacent sequences: A358502 A358503 A358504 * A358506 A358507 A358508

Keyword

nonn

Author

Gus Wiseman, Nov 20 2022

Status

approved