proposed

approved

editing

proposed

Cf. A001263, A004249, A005043, `A032027, A063895, A276625, A358373-A358377.

allocated for Gus WisemanNumbers k such that the k-th standard ordered rooted tree is fully canonically ordered (counted by A000081).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 21, 25, 27, 29, 31, 32, 37, 41, 43, 49, 53, 57, 59, 61, 63, 64, 65, 73, 81, 85, 101, 105, 107, 113, 117, 121, 123, 125, 127, 128, 129, 137, 145, 165, 169, 171, 193, 201, 209, 213, 229, 233, 235, 241, 245, 249, 251

1,2

The ordering of finitary multisets is first by length and then lexicographically. This is also the ordering used for Mathematica expressions.

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 corresponding ordered rooted trees begin:

1: o

2: (o)

3: ((o))

4: (oo)

5: (((o)))

7: (o(o))

8: (ooo)

9: ((oo))

11: ((o)(o))

13: (o((o)))

15: (oo(o))

16: (oooo)

17: ((((o))))

21: ((o)((o)))

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

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

Select[Range[1000], FreeQ[srt[#], _[__]?(!OrderedQ[#]&)]&]

These trees are counted by A000081.

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

Cf. A001263, A004249, A005043, `A032027, A063895, A276625, A358373-A358377.

allocated

nonn

Gus Wiseman, Nov 14 2022

approved

editing

allocated for Gus Wiseman

allocated

approved