A358457 - OEIS

%I #7 Nov 18 2022 23:36:56

%S 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 64,99,100,105,106,107,108,109,110,111,112,114,117,118,119,120,121,

%U 123,124,126,127,128,198,199,200,210,211,212,213,214,215,216,217,218

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

%C 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 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 terms together with their corresponding ordered trees begin:

%e    1: o

%e    2: (o)

%e    4: (oo)

%e    7: (o(o))

%e    8: (ooo)

%e   14: (o(o)o)

%e   15: (oo(o))

%e   16: (oooo)

%e   25: (o(oo))

%e   27: (o(o)(o))

%e   28: (o(o)oo)

%e   30: (oo(o)o)

%e   31: (ooo(o))

%e   32: (ooooo)

%e   50: (o(oo)o)

%e   53: (o(o)((o)))

%e   54: (o(o)(o)o)

%e   55: (o(o)o(o))

%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 Select[Range[100],Composition[Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]],srt]]

%Y The unordered version is A290822, counted by A290689.

%Y These trees are counted by A358453.

%Y The undirected version is A358458, counted by A358454.

%Y A000108 counts ordered rooted trees, unordered A000081.

%Y A306844 counts anti-transitive rooted trees.

%Y A324766 ranks recursively anti-transitive rooted trees, counted by A324765.

%Y A358455 counts recursively anti-transitive ordered rooted trees.

%Y Cf. A004249, A032027, A318185, A324695, A324758, A324766, A324840, A358373-A358377, A358456.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 18 2022