cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A127288 Signature-permutation of a Catalan automorphism: composition of A057508 and A127286.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 28, 30, 33, 24, 29, 26, 31, 32, 27, 34, 35, 36, 37, 39, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 70, 67, 72, 75, 79, 81
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This automorphism permutes the top-level of a list of even length (1 ... 2n) as (1 3 5 2n-1 ... 2n 2n-2 ... 4 2) and when applied to a list of odd length (1 .. 2n+1), permutes it as (1 3 5 2n-1 2n+1 2n 2n-2 ... 4 2).

Links

Crossrefs

Inverse: A127287. a(n) = A057508(A127286(n)).

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

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Scheme
    (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)))

Formula

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

A127285 Signature-permutation of a Catalan automorphism: SPINE-transformation of *A057508.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 10, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 37, 25, 42, 51, 24, 38, 26, 44, 47, 27, 53, 56, 60, 28, 39, 29, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 107, 70, 121, 149, 66
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

SPINE-transformation is explained in A122203. This automorphism permutes the top-level of a list of even length (1 ... 2n) as (2n 1 2n-1 2 2n-3 3 ... n+1 n) and when applied to a list of odd length (1 .. 2n+1), permutes it as (2n+1 1 2n 2 2n-1 3 ... n n+1). Used to construct A127287 and A127289.

Links

Crossrefs

Inverse: A127286. a(n) = A127287(A057508(n)).

A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 14, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 70, 72, 75, 67, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 ... 2n-1 2n 2n+1), permutes it as (2 1 4 3 ... 2n 2n-1 2n+1).

Links

Crossrefs

Cf. a(n) = A057508(A130373(A057508(n))) = A057164(A130373(A057164(n))) = A127285(A127288(n)) = A127287(A127286(n)). Also a(A085223(n)) = A130370(A122282(A130369(A085223(n)))) holds for all n>=0. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.
Showing 1-4 of 4 results.

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