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%I #8 Nov 21 2022 09:38:29
%S 0,2,12,10,56,50,44,42,52,226,204,202,184,178,172,170,240,210,908,906,
%T 824,818,812,810,180,738,716,714,696,690,684,682,228,962,844,842,3640,
%U 3634,3628,3626,820,3298,3276,3274,3256,3250,3244,3242,752,722,2956,2954
%N Binary encoding of the n-th standard ordered rooted tree.
%C 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.
%C 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.
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e The sixth standard tree is {{{}},{}}, which becomes (1,1,0,0,1,0), so a(6) = 50.
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t trt[t_]:=FromDigits[Take[DeleteCases[Characters[ToString[t]]/.{"{"->1,"}"->0},","|" "],{2,-2}],2];
%t Table[trt[srt[n]],{n,100}]
%Y Sorting gives A014486.
%Y A dual sequence is A358523.
%Y Cf. A000108, A001263, A057122, A358371, A358372, A358373, A358379, A358453.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 20 2022
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