cp's OEIS Frontend

A358523 Standard ordered tree numbers of ordered trees in order of their binary encodings (A014486).

%I A358523 #6 Nov 21 2022 09:38:25
%S A358523 1,2,4,3,8,7,6,9,5,16,15,14,25,13,12,11,18,129,65,10,33,257,17,32,31,
%T A358523 30,57,29,28,27,50,385,193,26,97,769,49,24,23,22,41,21,36,35,258,
%U A358523 32769,16385,130,8193,16777217,4097,20,19,66
%N A358523 Standard ordered tree numbers of ordered trees in order of their binary encodings (A014486).
%C A358523 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.
%C A358523 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.
%H A358523 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 A358523 The first six binary encodings are: 0, 2, 10, 12, 42, 44, and the corresponding trees have standard ranks: 1, 2, 4, 3, 8, 7.
%t A358523 stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
%t A358523 srtinv[t_]:=If[t=={},1,stcinv[srtinv/@t]+1];
%t A358523 binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
%t A358523 bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
%t A358523 Table[srtinv[bint[n]],{n,Select[Range[0,100],binbalQ]}]
%Y A358523 A dual sequence is A358505.
%Y A358523 A000108 counts ordered rooted trees, unordered A000081.
%Y A358523 A014486 lists all binary encodings.
%Y A358523 Cf. A001263, A014221, A057122, A358371, A358372, A358373, A358379.
%K A358523 nonn
%O A358523 0,2
%A A358523 _Gus Wiseman_, Nov 21 2022