A358506 -id:A358506 - cp's OEIS frontend

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

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

			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}]

A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 2, 6, 1, 1, 2, 2, 2, 6, 3, 2, 4, 4, 2, 6, 2, 3, 3, 2, 2, 5, 1, 3, 2, 6, 1, 4, 2, 4, 2, 4, 1, 12, 3, 2, 3, 1, 4, 6, 1, 3, 2, 6, 3, 10, 2, 6, 3, 3, 2, 12, 2, 5, 1, 4, 1, 12, 2, 4, 4, 4, 4, 12, 4, 3, 2, 4, 2, 6, 1, 3, 3, 6, 4, 6, 1, 8, 6, 2, 3, 10, 2, 6, 6, 5, 6, 6, 2, 6, 6, 2, 2, 20, 1, 6, 4, 3, 1, 12, 1, 1, 4, 12, 1, 12, 2, 2, 4, 4, 2, 6, 2, 12, 4, 6, 4, 15, 4, 4, 3, 9, 2, 12, 6, 4, 3, 6, 2, 24, 3, 4, 2, 6

Offset: 1

Emeric Deutsch, Apr 14 2012

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

a(n) = the number of times n occurs in A127301. - Antti Karttunen, Jan 03 2013

			a(4)=1 because the rooted tree with Matula number 4 is V and there is no other ordered tree isomorphic to V. a(6)=2 because the rooted tree corresponding to n = 6 is obtained by joining the trees A - B and C - D - E at their roots A and C. Interchanging their order, we obtain another ordered tree, isomorphic (as rooted tree) to the first one.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
P. Schultz, Enumeration of rooted trees with an application to group presentations, Discrete Math., 41, 1982, 199-214.
Index entries for sequences related to Matula-Goebel numbers

Cf. A127301.
Positions of 1's are 1 and A214577.
Positions of first appearances are A358507, unsorted A358508.
A000108 counts ordered rooted trees, unordered A000081.
A061775 and A196050 count nodes and edges in Matula-Goebel trees.
Cf. A001263, A003238, A014486, A358506.

with(numtheory): F := proc (n) options operator, arrow: factorset(n) end proc: PD := proc (n) local k, m, j: for k to nops(F(n)) do m[k] := 0: for j while is(n/F(n)[k]^j, integer) = true do m[k] := m[k]+1 end do end do: [seq([F(n)[q], m[q]], q = 1 .. nops(F(n)))] end proc: a := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n)) else mul(a(PD(n)[j][1])^PD(n)[j][2], j = 1 .. nops(F(n)))*factorial(add(PD(n)[k][2], k = 1 .. nops(F(n))))/mul(factorial(PD(n)[k][2]), k = 1 .. nops(F(n))) end if end proc: seq(a(n), n = 1 .. 160);

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}];
Table[treeperms[MGTree[n]],{n,100}] (* Gus Wiseman, Nov 21 2022 *)

a(1)=1; denoting by p(t) the t-th prime, if n = p(n_1)^{k_1}...p(n_r)^{k_r}, then a(n) = a(n_1)^{k_1}...a(n_r)^{k_r}*(k_1 + ... + k_r)!/[(k_1)!...(k_r)!] (see Theorem 1 in the Schultz reference, where the exponents k_j of N(n_j) have been inadvertently omitted).

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all 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

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 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}]]

cp's OEIS Frontend

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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A206487 Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.

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A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

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Mathematica

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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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