A358377 -id:A358377 - cp's OEIS frontend

A358508 Least Matula-Goebel number of a tree with exactly n permutations.

1, 6, 12, 24, 48, 30, 192, 104, 148, 72, 3072, 60, 12288, 832, 144, 712, 196608, 222, 786432, 120, 288, 13312

Offset: 1

Gus Wiseman, Nov 20 2022

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.

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

			The terms together with their corresponding trees begin:
      1: o
      6: (o(o))
     12: (oo(o))
     24: (ooo(o))
     48: (oooo(o))
     30: (o(o)((o)))
    192: (oooooo(o))
    104: (ooo(o(o)))
    148: (oo(oo(o)))
     72: (ooo(o)(o))
   3072: (oooooooooo(o))
     60: (oo(o)((o)))
  12288: (oooooooooooo(o))
    832: (oooooo(o(o)))
    144: (oooo(o)(o))
    712: (ooo(ooo(o)))

Position of first appearance of n in A206487.
The sorted version is A358507.
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}];
uv=Table[treeperms[MGTree[n]],{n,100000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]

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

Original entry on oeis.org

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}]]

A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 22, 24, 32, 33, 34, 35, 36, 37, 38, 40, 43, 44, 48, 64, 66, 67, 68, 69, 70, 72, 74, 75, 76, 80, 86, 88, 96, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 140, 144, 147, 148, 150, 152, 160, 171, 172

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

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.

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 standard ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: ((o)o)
   8: (ooo)
   9: ((oo))
  10: (((o))o)
  11: ((o)(o))
  12: ((o)oo)
  16: (oooo)
  17: ((((o))))
  18: ((oo)o)
  19: (((o))(o))
  20: (((o))oo)

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

Positions of first appearances in A358506.
The unsorted version is A358522.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[mgnum[srt[n]],{n,100}]]

A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 11, 10, 17, 12, 33, 18, 19, 16, 257, 22, 129, 20, 35, 34, 1025, 24, 37, 66, 43, 36, 513, 38, 65537, 32, 67, 514, 69, 44, 2049, 258, 131, 40

Offset: 1

Gus Wiseman, Nov 20 2022

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 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 standard ordered trees begin:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    6: ((o)o)
    9: ((oo))
    8: (ooo)
   11: ((o)(o))
   10: (((o))o)
   17: ((((o))))
   12: ((o)oo)
   33: (((o)o))
   18: ((oo)o)
   19: (((o))(o))
   16: (oooo)
  257: (((oo)))
   22: ((o)(o)o)
  129: ((ooo))
   20: (((o))oo)
   35: ((oo)(o))
   34: ((((o)))o)

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

Position of first appearance of n in A358506.
The sorted version is A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
uv=Table[mgnum[srt[n]],{n,10000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]

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]]

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 204, 212, 232, 240, 682, 820, 844, 852, 920, 936, 976, 992, 2730, 3276, 3284, 3380, 3404, 3412, 3640, 3688, 3736, 3752, 3888, 3920, 4000, 4032, 10922, 13108, 13132, 13140, 13516, 13524, 13620, 13644, 13652, 14568, 14744, 14760

Offset: 1

Gus Wiseman, Nov 21 2022

nonn

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

The binary encoding of an ordered tree (see A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

			The terms together with their corresponding trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  204: ((o)(o))
  212: ((ooo))
  232: (((oo)))
  240: ((((o))))
  682: (ooooo)
  820: ((o)(oo))
  844: ((oo)(o))
  852: ((oooo))
  920: (((o)(o)))
  936: (((ooo)))
  976: ((((oo))))
  992: (((((o)))))

These trees are counted by A007059.
This is a subset of A014486.
The version for binary trees is A057122.
The unordered version is A184155, counted by A048816.
Another ranking of balanced ordered trees is A358459.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A001263, A003238, A244925, A358377, A358379, A358523.

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
Select[Range[0,1000],binbalQ[#]&&SameQ@@Length/@Position[bint[#],{}]&]

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[#]]=={}&]

cp's OEIS Frontend

A358508 Least Matula-Goebel number of a tree with exactly n permutations.

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

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Mathematica

A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

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A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

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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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Mathematica

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

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Mathematica

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

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Mathematica