proposed

approved

editing

proposed

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

allocatedforGusWisemanNumbersksuchthatthek-thstandardorderedrootedtreeisfullycanonicallyordered(countedbyA000081).

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 /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 byA000081.

A358371andA358372count 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