A358376 - OEIS

%I #8 Nov 14 2022 20:00:52

%S 1,4,8,16,18,25,32,36,50,57,64,72,100,114,121,128,137,144,200,228,242,

%T 249,256,258,274,281,288,385,393,400,456,484,498,505,512,516,548,562,

%U 569,576,770,786,793,800,897,905,912,968,996,1010,1017,1024,1032,1096

%N Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

%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 initial terms and their corresponding trees:

%e     1: o

%e     4: (oo)

%e     8: (ooo)

%e    16: (oooo)

%e    18: ((oo)o)

%e    25: (o(oo))

%e    32: (ooooo)

%e    36: ((oo)oo)

%e    50: (o(oo)o)

%e    57: (oo(oo))

%e    64: (oooooo)

%e    72: ((oo)ooo)

%e   100: (o(oo)oo)

%e   114: (oo(oo)o)

%e   121: (ooo(oo))

%e   128: (ooooooo)

%e   137: ((oo)(oo))

%e   144: ((oo)oooo)

%e   200: (o(oo)ooo)

%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],FreeQ[srt[#],_[__]?(Length[#]==1&)]&]

%Y These trees are counted by A005043.

%Y The series-reduced case appears to be counted by A284778.

%Y The unordered version is A291636, counted by A001678.

%Y A000081 counts unlabeled rooted trees, ranked by A358378.

%Y A358371 and A358372 count leaves and nodes in standard ordered rooted trees.

%Y A358374 ranks ordered identity trees, counted by A032027.

%Y A358375 ranks ordered binary trees, counted by A126120.

%Y Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 14 2022