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approved

editing

proposed

22: ((o)(o)o)

24: ((o)ooo)

32: (ooooo)

33: (((o)o))

34: ((((o)))o)

35: ((oo)(o))

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

Cf. A001263, A014486, A061775, `A127301, A196050, A206487, A358371, A358372, ~`A358373, A358378, `A358379, ~`A358459, `A358505, ~`A358507, `~A358508, ~`A358523.

allocated for Gus WisemanSorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

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, 40, 43, 44, 48, 64, 66, 67, 68, 69, 70, 72, 74, 75, 76, 80, 86, 88, 96, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 140, 144, 147, 148, 150, 152, 160, 171, 172

1,2

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.

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.

Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

The terms together with their standard ordered trees begin:

1: o

2: (o)

3: ((o))

4: (oo)

5: (((o)))

6: ((o)o)

8: (ooo)

9: ((oo))

10: (((o))o)

11: ((o)(o))

12: ((o)oo)

16: (oooo)

17: ((((o))))

18: ((oo)o)

19: (((o))(o))

20: (((o))oo)

22: ((o)(o)o)

24: ((o)ooo)

32: (ooooo)

33: (((o)o))

34: ((((o)))o)

35: ((oo)(o))

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

srt[n_]:=If[n==1, {}, srt/@stc[n-1]];

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

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

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

Positions of first appearances in A358506.

The unsorted version is A358522.

A000108 counts ordered rooted trees, unordered A000081.

A214577 and A358377 rank trees with no permutations.

Cf. A001263, A014486, A061775, `A127301, A196050, A206487, A358371, A358372, ~`A358373, A358378, `A358379, ~`A358459, `A358505, ~`A358507, `~A358508, ~`A358523.

allocated

nonn

Gus Wiseman, Nov 20 2022

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allocated

allocated for Gus Wiseman

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