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A358457 Numbers k such that the k-th standard ordered rooted tree is transitive (counted by A358453).

%I A358457 #7 Nov 18 2022 23:36:56
%S A358457 1,2,4,7,8,14,15,16,25,27,28,30,31,32,50,53,54,55,56,57,59,60,62,63,
%T A358457 64,99,100,105,106,107,108,109,110,111,112,114,117,118,119,120,121,
%U A358457 123,124,126,127,128,198,199,200,210,211,212,213,214,215,216,217,218
%N A358457 Numbers k such that the k-th standard ordered rooted tree is transitive (counted by A358453).
%C A358457 We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root.
%C A358457 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 A358457 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 A358457 The terms together with their corresponding ordered trees begin:
%e A358457    1: o
%e A358457    2: (o)
%e A358457    4: (oo)
%e A358457    7: (o(o))
%e A358457    8: (ooo)
%e A358457   14: (o(o)o)
%e A358457   15: (oo(o))
%e A358457   16: (oooo)
%e A358457   25: (o(oo))
%e A358457   27: (o(o)(o))
%e A358457   28: (o(o)oo)
%e A358457   30: (oo(o)o)
%e A358457   31: (ooo(o))
%e A358457   32: (ooooo)
%e A358457   50: (o(oo)o)
%e A358457   53: (o(o)((o)))
%e A358457   54: (o(o)(o)o)
%e A358457   55: (o(o)o(o))
%t A358457 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358457 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358457 Select[Range[100],Composition[Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]],srt]]
%Y A358457 The unordered version is A290822, counted by A290689.
%Y A358457 These trees are counted by A358453.
%Y A358457 The undirected version is A358458, counted by A358454.
%Y A358457 A000108 counts ordered rooted trees, unordered A000081.
%Y A358457 A306844 counts anti-transitive rooted trees.
%Y A358457 A324766 ranks recursively anti-transitive rooted trees, counted by A324765.
%Y A358457 A358455 counts recursively anti-transitive ordered rooted trees.
%Y A358457 Cf. A004249, A032027, A318185, A324695, A324758, A324766, A324840, A358373-A358377, A358456.
%K A358457 nonn
%O A358457 1,2
%A A358457 _Gus Wiseman_, Nov 18 2022