A318185 -id:A318185 - cp's OEIS frontend

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

Gus Wiseman, Nov 18 2022

nonn

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.

			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))

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

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.

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]]

A358458 Numbers k such that the k-th standard ordered rooted tree is weakly transitive (counted by A358454).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 14, 15, 16, 18, 22, 23, 24, 25, 27, 28, 30, 31, 32, 36, 38, 39, 42, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 70, 71, 72, 76, 78, 79, 82, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 103, 105

Offset: 1

Gus Wiseman, Nov 18 2022

nonn

We define an unlabeled ordered rooted tree to be weakly transitive if every branch of a branch of the root is itself 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.

			The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   4: (oo)
   6: ((o)o)
   7: (o(o))
   8: (ooo)
  12: ((o)oo)
  14: (o(o)o)
  15: (oo(o))
  16: (oooo)
  18: ((oo)o)
  22: ((o)(o)o)
  23: ((o)o(o))
  24: ((o)ooo)

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

The unordered version is A290822, counted by A290689.
These trees are counted by A358454.
The directed version is A358457, counted by A358453.
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.

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],Complement[Union@@srt[#],srt[#]]=={}&]

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 301, 902, 2767, 8637, 27324, 87409, 282319, 919352

Offset: 1

Gus Wiseman, Nov 19 2022

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex.

			The a(1) = 1 through a(5) = 13 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

The locally non-intersecting version is A143363, unordered A007562.
The unordered version is A316473, ranked by A316495.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A006013, A302696, A316471, A316694, A318185, A319378, A324768, A324844.

aot[n_]:=If[n==1,{{}},Join @@ Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,{_,x_,_},_,{_,x_,_},_}]&]],{n,10}]

A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563

Offset: 1

Gus Wiseman, Nov 19 2022

nonn

An ordered tree is balanced if all leaves have the same distance from 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.

			The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))

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

These trees are counted by A007059.
The unordered version is A184155, counted by A048816.
A000108 counts ordered rooted trees, unordered A000081.
A358379 gives depth of standard ordered trees.
Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

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],SameQ@@Length/@Position[srt[#],{}]&]

cp's OEIS Frontend

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

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A358458 Numbers k such that the k-th standard ordered rooted tree is weakly transitive (counted by A358454).

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A358460 Number of locally disjoint ordered rooted trees with n nodes.

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A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

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Mathematica