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

Author

Gus Wiseman, Nov 20 2022

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Sorting gives A014486.

A dual sequence is A358523.

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

Programs

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}]