%I #7 Nov 20 2022 18:30:21

%S 1,2,3,4,5,6,8,9,10,11,12,16,17,18,19,20,22,24,32,33,34,35,36,37,38,

%T 40,43,44,48,64,66,67,68,69,70,72,74,75,76,80,86,88,96,128,129,131,

%U 132,133,134,136,137,138,139,140,144,147,148,150,152,160,171,172

%N Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

%C The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

%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 standard ordered trees begin:

%e 1: o

%e 2: (o)

%e 3: ((o))

%e 4: (oo)

%e 5: (((o)))

%e 6: ((o)o)

%e 8: (ooo)

%e 9: ((oo))

%e 10: (((o))o)

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

%e 12: ((o)oo)

%e 16: (oooo)

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

%e 18: ((oo)o)

%e 19: (((o))(o))

%e 20: (((o))oo)

%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 mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];

%t fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];

%t fir[Table[mgnum[srt[n]],{n,100}]]

%Y Positions of first appearances in A358506.

%Y The unsorted version is A358522.

%Y A000108 counts ordered rooted trees, unordered A000081.

%Y A214577 and A358377 rank trees with no permutations.

%Y Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 20 2022