A032027 -id:A032027 - cp's OEIS frontend

A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

1, 6, 12, 24, 30, 48, 60, 72, 104, 120, 144, 148, 156, 180, 192, 222, 288, 312, 360, 390, 432, 444, 480, 576, 712, 720, 780, 832, 864, 900, 1080, 1110, 1248, 1260, 1296, 1440, 1560, 1680, 2136, 2160, 2262, 2304, 2340, 2496, 2520, 2592, 2738, 2880, 2886, 3072

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding trees begin:
    1: o
    6: (o(o))
   12: (oo(o))
   24: (ooo(o))
   30: (o(o)((o)))
   48: (oooo(o))
   60: (oo(o)((o)))
   72: (ooo(o)(o))
  104: (ooo(o(o)))
  120: (ooo(o)((o)))
  144: (oooo(o)(o))
  148: (oo(oo(o)))
  156: (oo(o)(o(o)))
  180: (oo(o)(o)((o)))
  192: (oooooo(o))
  222: (o(o)(oo(o)))
  288: (ooooo(o)(o))
  312: (ooo(o)(o(o)))

Positions of first appearances in A206487.
The unsorted version is A358508.
A000081 counts rooted trees, ordered A000108.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
treeperms[t_]:=Times@@Cases[t,b:{}:>Length[Permutations[b]],{0,Infinity}];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[treeperms[MGTree[n]],{n,100}]]

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[#]]=={}&]

A032028 Number of rooted planar trees (n+1 nodes) where any 2 subtrees extending from same node are different.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 23, 62, 167, 461, 1309, 3784, 11018, 32431, 96294, 288246, 868107, 2629499, 8005166, 24482202, 75177313, 231697259, 716488213, 2222381516, 6912569477, 21556270329, 67380727837, 211080434346

Offset: 1

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to rooted trees

AGK(v)={apply(p->subst(serlaplace(y^0*p),y,1), Vec(prod(k=1, #v, (1 + x^k*y + O(x*x^#v))^v[k])-1, -#v))}
CGK(v)={apply(p->subst(serlaplace(p/y),y,1), Vec(prod(k=1, #v, (1 + x^k*y + O(x*x^#v))^v[k])-1, -#v))}
seq(n)={my(v=[1]); for(i=3, n, v=concat([1], AGK(v))); concat([1], CGK(v))} \\ Andrew Howroyd, Sep 20 2018

"CGK" (necklace, elements, unlabeled) transform of A032027 (shifted right one place).

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

A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

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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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Mathematica

A032028 Number of rooted planar trees (n+1 nodes) where any 2 subtrees extending from same node are different.

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PARI

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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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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica