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A358458 Numbers k such that the k-th standard ordered rooted tree is weakly transitive (counted by A358454).

%I A358458 #5 Nov 18 2022 23:36:40
%S A358458 1,2,4,6,7,8,12,14,15,16,18,22,23,24,25,27,28,30,31,32,36,38,39,42,44,
%T A358458 45,46,47,48,50,51,53,54,55,56,57,59,60,62,63,64,70,71,72,76,78,79,82,
%U A358458 84,86,87,88,89,90,91,92,93,94,95,96,99,100,102,103,105
%N A358458 Numbers k such that the k-th standard ordered rooted tree is weakly transitive (counted by A358454).
%C A358458 We define an unlabeled ordered rooted tree to be weakly transitive if every branch of a branch of the root is itself a branch of the root.
%C A358458 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 A358458 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 A358458 The terms together with their corresponding ordered trees begin:
%e A358458    1: o
%e A358458    2: (o)
%e A358458    4: (oo)
%e A358458    6: ((o)o)
%e A358458    7: (o(o))
%e A358458    8: (ooo)
%e A358458   12: ((o)oo)
%e A358458   14: (o(o)o)
%e A358458   15: (oo(o))
%e A358458   16: (oooo)
%e A358458   18: ((oo)o)
%e A358458   22: ((o)(o)o)
%e A358458   23: ((o)o(o))
%e A358458   24: ((o)ooo)
%t A358458 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358458 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358458 Select[Range[100],Complement[Union@@srt[#],srt[#]]=={}&]
%Y A358458 The unordered version is A290822, counted by A290689.
%Y A358458 These trees are counted by A358454.
%Y A358458 The directed version is A358457, counted by A358453.
%Y A358458 A000108 counts ordered rooted trees, unordered A000081.
%Y A358458 A306844 counts anti-transitive rooted trees.
%Y A358458 A324766 ranks recursively anti-transitive rooted trees, counted by A324765.
%Y A358458 A358455 counts recursively anti-transitive ordered rooted trees.
%Y A358458 Cf. A004249, A032027, A318185, A324695, A324758, A324766, A324840, A358373-A358377, A358456.
%K A358458 nonn
%O A358458 1,2
%A A358458 _Gus Wiseman_, Nov 18 2022