A358459 - OEIS

%I #6 Nov 19 2022 08:56:49

%S 1,2,3,4,5,8,9,11,16,17,32,35,37,41,43,64,128,129,137,139,163,169,171,

%T 256,257,293,512,515,529,547,553,555,641,649,651,675,681,683,1024,

%U 1025,2048,2053,2057,2059,2177,2185,2187,2211,2217,2219,2305,2341,2563

%N Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

%C An ordered tree is balanced if all leaves have the same distance from 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    3: ((o))

%e    4: (oo)

%e    5: (((o)))

%e    8: (ooo)

%e    9: ((oo))

%e   11: ((o)(o))

%e   16: (oooo)

%e   17: ((((o))))

%e   32: (ooooo)

%e   35: ((oo)(o))

%e   37: (((o))((o)))

%e   41: ((o)(oo))

%e   43: ((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],SameQ@@Length/@Position[srt[#],{}]&]

%Y These trees are counted by A007059.

%Y The unordered version is A184155, counted by A048816.

%Y A000108 counts ordered rooted trees, unordered A000081.

%Y A358379 gives depth of standard ordered trees.

%Y Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 19 2022