A324840 -id:A324840 - cp's OEIS frontend

A358454 Number of weakly transitive ordered rooted trees with n nodes.

1, 1, 1, 3, 6, 13, 33, 80, 201, 509, 1330, 3432, 8982, 23559, 62189

Offset: 1

Gus Wiseman, Nov 18 2022

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.

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((o)o)  ((o)oo)  ((o)ooo)
                (o(o))  ((oo)o)  ((oo)oo)
                        (o(o)o)  ((ooo)o)
                        (o(oo))  (o(o)oo)
                        (oo(o))  (o(oo)o)
                                 (o(ooo))
                                 (oo(o)o)
                                 (oo(oo))
                                 (ooo(o))
                                 ((o)(o)o)
                                 ((o)o(o))
                                 (o(o)(o))

The unordered version is A290689, ranked by A290822.
The directed version is A358453.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

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

A358455 Number of recursively anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 72, 206, 608, 1830, 5612, 17442, 54866, 174252, 558072, 1800098

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be recursively anti-transitive if no branch of a branch of a subtree is a branch of the same subtree farther to the left.

			The a(1) = 1 through a(5) = 10 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (((o)))  ((ooo))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o)(o))
                          (o((o)))
                          ((((o))))

The unordered version is A324765, ranked by A324766.
The undirected version is A358456.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, 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}]

A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 35, 49, 51, 53, 57, 59, 63, 64, 67, 73, 77, 81, 83, 85, 95, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 149, 153, 159, 161, 171, 175, 177, 187, 189, 201, 209, 217, 227, 233, 241, 243, 245

Offset: 1

Gus Wiseman, Mar 17 2019

nonn

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

			The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  35: (((o))(oo))
  49: ((oo)(oo))
  51: ((o)((oo)))
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  63: ((o)(o)(oo))

Cf. A000081, A007097, A290689, A303431, A304360, A306844, A316502, A318185, A318186.
Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324769.
Cf. A324838, A324840, A324844.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={},And@@fratQ/@primeMS[n]];
Select[Range[100],fratQ]

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

Original entry on oeis.org

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[#]]=={}&]

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A358454 Number of weakly transitive ordered rooted trees with n nodes.

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A358455 Number of recursively anti-transitive ordered rooted trees with n nodes.

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A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

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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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