A358371 -id:A358371 - cp's OEIS frontend

A358378 Numbers k such that the k-th standard ordered rooted tree is fully canonically ordered (counted by A000081).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 21, 25, 27, 29, 31, 32, 37, 41, 43, 49, 53, 57, 59, 61, 63, 64, 65, 73, 81, 85, 101, 105, 107, 113, 117, 121, 123, 125, 127, 128, 129, 137, 145, 165, 169, 171, 193, 201, 209, 213, 229, 233, 235, 241, 245, 249, 251

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

The ordering of finitary multisets is first by length and then lexicographically. This is also the ordering used for Mathematica expressions.

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 rooted trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: (o(o))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  13: (o((o)))
  15: (oo(o))
  16: (oooo)
  17: ((((o))))
  21: ((o)((o)))

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

These trees are counted by A000081.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
Cf. A001263, A004249, A005043, A032027, A063895, A276625, A358373-A358377.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(!OrderedQ[#]&)]&]

A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 9, 20, 22, 23, 26, 27, 29, 35, 41, 66, 76, 78, 79, 84, 86, 87, 90, 91, 93, 97, 102, 103, 106, 107, 109, 115, 117, 130, 136, 138, 139, 141, 146, 153, 163, 169, 193, 196, 197, 201, 226, 241, 262, 263, 296, 300, 302, 303, 308, 310, 311, 314, 315, 317

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

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 rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
   9: ((o)(o))
  20: (oo((o)))
  22: (o(((o))))
  23: (((o)(o)))
  26: (o(o(o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  35: (((o))(oo))
  41: (((o(o))))
  66: (o(o)(((o))))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  79: ((o(((o)))))
  84: (oo(o)(oo))
  86: (o(o(oo)))

These ordered trees are counted by A000891.
The unordered version is A358578, counted by A185650.
Height instead of leaves: counted by A358588, unordered A358576.
Height instead of internals: counted by A358590, unordered A358577.
Standard ordered tree number statistics: A358371, A358372, A358379, A358553.
A000081 counts rooted trees, ordered A000108.
A055277 counts trees by nodes and leaves, ordered A001263.
Cf. A014221, A206487, A358373, A358580, A358587, A358589.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],Count[srt[#],{},{0,Infinity}]==Count[srt[#],[_],{0,Infinity}]&]

A358371(a(n)) = A358553(a(n)).

A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 41, 171, 633, 2171, 7070, 22195, 67830, 203130, 598806, 1743258, 5023711, 14356226, 40737383, 114904941, 322432215, 900707165, 2506181060, 6948996085, 19207795836, 52944197508, 145567226556, 399314965956, 1093107693133, 2986640695436

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

			The a(5) = 1 and a(6) = 8 ordered trees:
  ((o)(o))  ((o)(o)o)
            ((o)(oo))
            ((o)o(o))
            ((oo)(o))
            (o(o)(o))
            (((o))(o))
            (((o)(o)))
            ((o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For leaves instead of height we have A000891, unordered A185650 aerated.
The unordered version is A358587, ranked by A358576.
For leaves instead of internal nodes we have A358590, unordered A358589.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
Cf. A065097, A342507, A358552, A358371, A358578, A358579, A358586, A358591.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,[_],{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ Needs R(n,f) defined in A358590.
seq(n) = {Vec(R(n, (h,p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Conjectures from Chai Wah Wu, Apr 14 2024: (Start)
a(n) = 9*a(n-1) - 32*a(n-2) + 58*a(n-3) - 58*a(n-4) + 32*a(n-5) - 9*a(n-6) + a(n-7) for n > 7.
G.f.: x^5*(-x^2 + x - 1)/((x - 1)^3*(x^2 - 3*x + 1)^2). (End)

Terms a(16) and beyond from Andrew Howroyd, Jan 01 2023

A358505 Binary encoding of the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 2, 12, 10, 56, 50, 44, 42, 52, 226, 204, 202, 184, 178, 172, 170, 240, 210, 908, 906, 824, 818, 812, 810, 180, 738, 716, 714, 696, 690, 684, 682, 228, 962, 844, 842, 3640, 3634, 3628, 3626, 820, 3298, 3276, 3274, 3256, 3250, 3244, 3242, 752, 722, 2956, 2954

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

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

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 sixth standard tree is {{{}},{}}, which becomes (1,1,0,0,1,0), so a(6) = 50.

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

Sorting gives A014486.
A dual sequence is A358523.
Cf. A000108, A001263, A057122, A358371, A358372, A358373, A358379, A358453.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
trt[t_]:=FromDigits[Take[DeleteCases[Characters[ToString[t]]/.{"{"->1,"}"->0},","|" "],{2,-2}],2];
Table[trt[srt[n]],{n,100}]

A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 7, 11, 66, 127, 715, 1549, 8398, 19691, 104006, 258194, 1337220, 3467115, 17678835, 47440745, 238819350, 659060677, 3282060210, 9271024542, 45741281820, 131788178171, 644952073662, 1890110798926, 9183676536076, 27316119923002, 131873975875180, 397407983278484

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(6) = 11 ordered trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)
                     ((o)oo)  ((o)ooo)
                     ((oo)o)  ((oo)oo)
                     ((ooo))  ((ooo)o)
                     (o(o)o)  ((oooo))
                     (o(oo))  (o(o)oo)
                     (oo(o))  (o(oo)o)
                              (o(ooo))
                              (oo(o)o)
                              (oo(oo))
                              (ooo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

For equality we have A000891, unordered A185650.
Odd-indexed terms are A065097.
The unordered version is A358581.
The opposite is the same, unordered A358582.
The non-strict version is A358586, unordered A358583.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358584, A358588, A358590.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]>Count[#,[_],{0,Infinity}]&]],{n,10}]

a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) - if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor((n-1)/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) - A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

a(16) onwards from Andrew Howroyd, Jan 13 2024

A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

Original entry on oeis.org

1, 4, 8, 16, 18, 25, 32, 36, 50, 57, 64, 72, 100, 114, 121, 128, 137, 144, 200, 228, 242, 249, 256, 258, 274, 281, 288, 385, 393, 400, 456, 484, 498, 505, 512, 516, 548, 562, 569, 576, 770, 786, 793, 800, 897, 905, 912, 968, 996, 1010, 1017, 1024, 1032, 1096

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

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 initial terms and their corresponding trees:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   18: ((oo)o)
   25: (o(oo))
   32: (ooooo)
   36: ((oo)oo)
   50: (o(oo)o)
   57: (oo(oo))
   64: (oooooo)
   72: ((oo)ooo)
  100: (o(oo)oo)
  114: (oo(oo)o)
  121: (ooo(oo))
  128: (ooooooo)
  137: ((oo)(oo))
  144: ((oo)oooo)
  200: (o(oo)ooo)

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

These trees are counted by A005043.
The series-reduced case appears to be counted by A284778.
The unordered version is A291636, counted by A001678.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120.
Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

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[#],[_]?(Length[#]==1&)]&]

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

Original entry on oeis.org

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

Gus Wiseman, Nov 14 2022

nonn

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.

			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)))

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

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.

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@@#&)]&]

A358375 Numbers k such that the k-th standard ordered rooted tree is binary.

Original entry on oeis.org

1, 4, 18, 25, 137, 262146, 393217, 2097161, 2228225

Offset: 1

Gus Wiseman, Nov 14 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 initial terms and their corresponding trees:
       1: o
       4: (oo)
      18: ((oo)o)
      25: (o(oo))
     137: ((oo)(oo))
  262146: (((oo)o)o)
  393217: (o((oo)o))

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

The unordered version is A111299, counted by A001190
These trees are counted by A126120.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
Cf. A001263, A004249, A005043, A063895, A245824, A284778, A358373, A358374, A358376, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(Length[#]!=2&)]&]

A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 13, 26, 33, 44

Offset: 1

Gus Wiseman, Nov 20 2022

nonn

First differs from A333219 at a(65) = 13, A333219(65) = 17.
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 first eight standard ordered trees are: o, (o), ((o)), (oo), (((o))), ((o)o), (o(o)), (ooo), with Matula-Goebel numbers: 1, 2, 3, 4, 5, 6, 6, 8.

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

For binary instead of standard encoding we have A127301.
There are exactly A206487(n) appearances of n.
For binary instead of Matula-Goebel encoding we have A358505.
Positions of first appearances are A358522, sorted A358521.
A000108 counts ordered rooted trees, unordered A000081.
A214577 and A358377 rank trees with no permutations.
Cf. A001263, A014486, A061775, A196050, A358371, A358372, A358378, A358379, A358507, A358508.

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];
Table[mgnum[srt[n]],{n,100}]

A358523 Standard ordered tree numbers of ordered trees in order of their binary encodings (A014486).

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 6, 9, 5, 16, 15, 14, 25, 13, 12, 11, 18, 129, 65, 10, 33, 257, 17, 32, 31, 30, 57, 29, 28, 27, 50, 385, 193, 26, 97, 769, 49, 24, 23, 22, 41, 21, 36, 35, 258, 32769, 16385, 130, 8193, 16777217, 4097, 20, 19, 66

Offset: 0

Gus Wiseman, Nov 21 2022

nonn

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

			The first six binary encodings are: 0, 2, 10, 12, 42, 44, and the corresponding trees have standard ranks: 1, 2, 4, 3, 8, 7.

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

A dual sequence is A358505.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263, A014221, A057122, A358371, A358372, A358373, A358379.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
srtinv[t_]:=If[t=={},1,stcinv[srtinv/@t]+1];
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->"}"}],","->""],"} {"->"},{"]]]
Table[srtinv[bint[n]],{n,Select[Range[0,100],binbalQ]}]

cp's OEIS Frontend

A358378 Numbers k such that the k-th standard ordered rooted tree is fully canonically ordered (counted by A000081).

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A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

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A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

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A358505 Binary encoding of the n-th standard ordered rooted tree.

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A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

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A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

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

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A358375 Numbers k such that the k-th standard ordered rooted tree is binary.

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