A358457 Numbers k such that the k-th standard ordered rooted tree is transitive (counted by A358453).

1, 2, 4, 7, 8, 14, 15, 16, 25, 27, 28, 30, 31, 32, 50, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 99, 100, 105, 106, 107, 108, 109, 110, 111, 112, 114, 117, 118, 119, 120, 121, 123, 124, 126, 127, 128, 198, 199, 200, 210, 211, 212, 213, 214, 215, 216, 217, 218

Offset

1,2

Comments

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root.

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.

Links

Table of n, a(n) for n=1..58.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Example

The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   4: (oo)
   7: (o(o))
   8: (ooo)
  14: (o(o)o)
  15: (oo(o))
  16: (oooo)
  25: (o(oo))
  27: (o(o)(o))
  28: (o(o)oo)
  30: (oo(o)o)
  31: (ooo(o))
  32: (ooooo)
  50: (o(oo)o)
  53: (o(o)((o)))
  54: (o(o)(o)o)
  55: (o(o)o(o))

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Select[Range[100], Composition[Function[t, And@@Table[Complement[t[[k]], Take[t, k]]=={}, {k, Length[t]}]], srt]]

Crossrefs

The unordered version is A290822, counted by A290689.

These trees are counted by A358453.

The undirected version is A358458, counted by A358454.

A000108 counts ordered rooted trees, unordered A000081.

A306844 counts anti-transitive rooted trees.

A324766 ranks recursively anti-transitive rooted trees, counted by A324765.

A358455 counts recursively anti-transitive ordered rooted trees.

Cf. A004249, A032027, A318185, A324695, A324758, A324766, A324840, A358373-A358377, A358456.

Sequence in context: A287208 A212243 A336463 * A306235 A353441 A004616

Adjacent sequences: A358454 A358455 A358456 * A358458 A358459 A358460

Keyword

nonn

Author

Gus Wiseman, Nov 18 2022

Status

approved