A358374 Numbers k such that the k-th standard ordered rooted tree is an identity tree (counted by A032027).

1, 2, 3, 5, 6, 7, 10, 13, 17, 19, 21, 33, 34, 38, 39, 42, 45, 49, 51, 53, 65, 66, 67, 81, 97, 130, 131, 133, 134, 135, 145, 161, 162, 177, 193, 195, 209, 259, 261, 262, 263, 266, 269, 289, 290, 305, 321, 322, 353, 387, 389, 401, 417, 513, 517, 518, 519, 522

Offset: 1

Author

Gus Wiseman, Nov 14 2022

Keywords

• nonn

Comments

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.

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

Examples

			The terms together with their corresponding ordered rooted trees begin:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: ((o)o)
   7: (o(o))
  10: (((o))o)
  13: (o((o)))
  17: ((((o))))
  19: (((o))(o))
  21: ((o)((o)))
  33: (((o)o))
  34: ((((o)))o)
  38: (((o))(o)o)
  39: (((o))o(o))
  42: ((o)((o))o)
  45: ((o)o((o)))
		

Links

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

Crossrefs

These trees are counted by A032027.

The unordered version is A276625, counted by A004111.

A000081 counts unlabeled rooted trees, ranked by A358378.

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

A358375 ranks ordered binary trees, counted by A126120.

Cf. A001263, A004249, A005043, A063895, A358373, A358376, A358377.

Programs

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],FreeQ[srt[#],[_]?(!UnsameQ@@#&)]&]