A127301 -id:A127301 - cp's OEIS frontend

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Index entries for sequences related to Matula-Goebel numbers

Cf A127301, A153835, A153829.

a(n) = A127301(A057123(n)).

Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A129593 Prime-factorization encoded partition codes for the Łukasiewicz-words in A071153.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 9, 9, 5, 16, 27, 27, 6, 25, 27, 25, 6, 27, 25, 25, 25, 25, 7, 32, 81, 81, 18, 125, 81, 125, 18, 18, 15, 125, 15, 15, 49, 81, 125, 125, 15, 49, 18, 15, 18, 81, 125, 15, 125, 15, 49, 125, 49, 15, 125, 49, 15, 15, 125, 49, 49, 49, 49, 49, 11, 64, 243, 243, 54

Offset: 0

Antti Karttunen, May 01 2007

nonn

If the signature-permutation of a Catalan automorphism SP satisfies the condition A129593(SP(n)) = A129593(n) for all n, then it is called a Łukasiewicz-word permuting automorphism. In addition to all the automorphisms whose signature permutation satisfies the more restricted condition A127301(SP(n)) = A127301(n) for all n, this includes also certain automorphisms like *A072797 that do not preserve the non-oriented form of the general tree. A000041(n) distinct values occur in each range [A014137(n-1)..A014138(n-1)]. All natural numbers occur. Cf. A129599.

			The terms A071153(5..7) are 201, 210 and 120. After discarding zero and sorting, each produces partition 1+2. Converting it to prime-exponents like explained in A129595, we get 2^0 * 3^2 = 9, thus a(5) = a(6) = a(7) = 9.

A. Karttunen, Table of n, a(n) for n = 0..2055
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz

a(n) = a(A072797(n)).

Variant: A129599. To be computed: the position of the first and the last occurrence of n, the number of occurrences of each n.

Construction: remove zeros from the Łukasiewicz-word of a general plane tree encoded by A014486(n) (i.e. A071153(n)), sort the numbers into ascending order and interpreting it as a partition of a natural number, encode it in the manner explained in A129595.

A209636 Matula-numbers computed for rooted trees encoded by A071162/A071163.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 5, 16, 12, 14, 10, 19, 13, 17, 11, 32, 24, 28, 20, 38, 26, 34, 22, 53, 37, 43, 29, 67, 41, 59, 31, 64, 48, 56, 40, 76, 52, 68, 44, 106, 74, 86, 58, 134, 82, 118, 62, 131, 89, 107, 71, 163, 101, 139, 79, 241, 157, 191, 109, 331, 179, 277

Offset: 0

Antti Karttunen, Mar 11 2012

Sequence is injective: Any number occurs at most once, as each plane tree encoded by A071162/A071163 is mapped to a unique non-oriented rooted tree. See also A209637, A209638.

Sequence A209638 gives the same terms sorted into ascending order.

Indranil Ghosh (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Matula-Goebel numbers

Cf. A000040, A054429, A071162, A071163, A127301, A209637, A278541.

A209636(n) = { my(n=2*n, m=1); while(n >= 2, if(!(n%2),m*=2,m = prime(m)); n\=2); m; } \\ Antti Karttunen, May 25 2017

from sympy import prime
def a(n):
    n = 2*n
    m = 1
    if n<2: return 1
    while n>1:
        if n%2==0:
            n//=2
            m*=2
        else:
            n=(n - 1)//2
            m=prime(m)
    return m
print([a(n) for n in range(101)]) # Indranil Ghosh, May 25 2017, translated from Antti Karttunen's SCHEME code

(define (A209636 n) (let loop ((n (* 2 n)) (m 1)) (cond ((< n 2) m) ((even? n) (loop (/ n 2) (* m 2))) (else (loop (/ (- n 1) 2) (A000040 m))))))

a(n) = A127301(A071163(n)) = A209637(A054429(n)).

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}]

A153828 Index sequence to A089840: set-wise difference of A153827 and A153826.

Original entry on oeis.org

8, 45, 71, 115, 119, 121, 125, 127, 396, 397, 398, 399, 514, 525, 526, 532, 633, 635, 636, 637, 656, 657, 658, 659, 660, 661, 752, 757, 758, 874, 880, 888, 892, 993, 1001, 1120, 1121, 1126, 1127, 1156, 1157, 1168, 1169, 1174, 1175, 1347, 1394, 1395

Offset: 0

Antti Karttunen, Jan 07 2009

nonn

The terms give the positions of bijections in A089840 which preserve A129593, but not A127301.

A. Karttunen, Table of n, a(n) for n = 0..9292
A. Karttunen, C-program for computing the initial terms of this sequence

Cf. also A153829, A153830, A153831, A153832, A153833.

A209637 Matula-numbers computed for rooted trees encoded by A071162 when interpreted in once-halved bit-tuple format.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 11, 17, 13, 19, 10, 14, 12, 16, 31, 59, 41, 67, 29, 43, 37, 53, 22, 34, 26, 38, 20, 28, 24, 32, 127, 277, 179, 331, 109, 191, 157, 241, 79, 139, 101, 163, 71, 107, 89, 131, 62, 118, 82, 134, 58, 86, 74, 106, 44, 68, 52, 76, 40, 56, 48

Offset: 0

Antti Karttunen, Mar 11 2012

Sequence A209638 gives the same terms sorted into ascending order.

Mueller, Szymanski, Knop and Trinajstic, A Comparison between the Matula Numbers and Bit-tuple Notation for Rooted Trees J. Chem. Inf. Comput. Sci. 1995, 35, pp. 211--213.

from sympy import prime
from mpmath import log
def a054429(n): return 3*(2**int(log(n, 2))) - (n + 1)
def a209636(n):
    n = 2*n
    m = 1
    if n<2: return 1
    while n>1:
        if n%2==0:
            n/=2
            m*=2
        else:
            n=(n - 1)/2
            m=prime(m)
    return m
def a(n): return 1 if n==0 else a209636(a054429(n)) # Indranil Ghosh, May 26 2017

a(n) = A209636(A054429(n)) = A127301(A057505(A071163(n))) = A127301(A057163(A071163(n))).

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

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A129593 Prime-factorization encoded partition codes for the Łukasiewicz-words in A071153.

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A209636 Matula-numbers computed for rooted trees encoded by A071162/A071163.

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PARI

Python

Scheme

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

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Mathematica

A153828 Index sequence to A089840: set-wise difference of A153827 and A153826.

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A209637 Matula-numbers computed for rooted trees encoded by A071162 when interpreted in once-halved bit-tuple format.

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Python

Formula

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

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Mathematica

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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