A153835 -id:A153835 - cp's OEIS frontend

A153829 Index sequence to A089840: positions of bijections that preserve A153835, or equivalently, A127302 (the non-oriented form of binary trees).

0, 1, 3, 7, 15, 21, 27, 46, 68, 73, 74, 83, 84, 87, 88, 92, 114, 149, 169, 183, 184, 189, 190, 199, 202, 203, 225, 251, 252, 254, 261, 262, 268, 269, 270, 271, 299, 400, 515, 537, 539, 573, 575, 591, 593, 638, 753, 871, 894, 895, 990, 995, 996, 1110, 1132

Offset: 0

Antti Karttunen, Jan 07 2009

nonn

These elements form a subgroup in A089840 (A089839). Such elements consists of only such clauses where each vertex stays at the same distance from the root of the binary tree and in the image tree will still be sibling to its original sibling in the pre-image tree.

Because A127302 can be computed as a fold and most of the recursive derivations of A089840 (i.e. tables A122201-A122204, A122283-A122290, A130400-A130403) are also folds, this sequence gives also the indices to those derived tables where bijections preserving A127302 occur.

A. Karttunen, Table of n, a(n) for n = 0..3109
A. Karttunen, C-program for computing the initial terms of this sequence

Superset of A153830. Apart from 0, has no other elements common with A153826. Cf. also A153831, A153827, A153829, A153832, A153833.

A154103 Auxiliary array for computing A153835.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

Antti Karttunen, Jan 05 2009

A(x,y): A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... is 1 if y=0 and x!=0, or if A085207bi(2*x, y) is greater than A085207bi(2*y, x), 0 otherwise.

A. Karttunen, Table of n, a(n) for n = 0..1274

Transpose: A154104. Cf. A085207, A002262, A025581.

A069770 Signature permutation of the first non-identity, nonrecursive Catalan automorphism in table A089840: swap the top branches of a binary tree. An involution of nonnegative integers.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 4, 5, 17, 18, 20, 21, 22, 16, 19, 14, 9, 10, 15, 11, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 129, 130, 132, 133, 134

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

This is the simplest possible Catalan automorphism after the identity bijection (A001477). It effects the following transformation on the unlabeled rooted plane binary trees (letters A and B refer to arbitrary subtrees located on those vectices):
   A   B           B   A
    \ /     -->     \ /
     x               x
  (a . b)  -----> (b . a)
Applying this permutation recursively to the right hand side branch of the binary trees produces permutations A069767 and A069768 (that occur at the same index 1 in tables A122203 and A122204), and applying this recursively to the both branches of binary trees (as in pre- or postorder traversal) produces A057163 (which occurs at the same index 1 in tables A122201 and A122202) that reflects the whole binary tree.
For this permutation, A127302(a(n)) = A127302(n) for all n, [or equally, A153835(a(n)) = A153835(n)], and likewise for all such recursive derivations as mentioned above.

			To obtain the signature permutation, we apply these transformations to the binary trees as encoded and ordered by A014486 and for each n, a(n) will be the position of the tree to which the n-th tree is transformed to, as follows:
.
                   one tree of one internal
  empty tree         (non-leaf) node
      x                      \/
n=    0                      1
a(n)= 0                      1               (both are always fixed)
.
the next 7 trees, with 2-3 internal nodes, in range [A014137(1), A014137(2+1)-1] = [2,8] are:
.
                          \/     \/                 \/     \/
       \/     \/         \/       \/     \/ \/     \/       \/
      \/       \/       \/       \/       \_/       \/       \/
n=     2        3        4        5        6        7        8
.
and the new shapes after swapping their left and right hand subtrees are:
.
                        \/     \/                     \/     \/
     \/         \/     \/       \/       \/ \/       \/       \/
      \/       \/       \/       \/       \_/       \/       \/
a(n)=  3        2        7        8        6        4        5
thus we obtain the first nine terms of this sequence: 0, 1, 3, 2, 7, 8, 6, 4, 5.

Row 1 of A089840.
The number of cycles and the number of fixed points in each subrange limited by terms of A014137 are given by A007595 and A097331.
Other related sequences: A014486, A057163, A069767, A069768, A089864, A123492, A154125, A154126.
Cf. also A127302, A153835.

a(n) = A154125(A154126(n)) = A154126(A154125(n)).

a(n) = A083927(A072796(A057123(n))) = A083927(A057508(A057123(n))) = A083927(A057509(A057123(n))).

Entry revised by Antti Karttunen, Oct 11 2006 and Mar 30 2024

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

Original entry on oeis.org

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

A153831 Index sequence to A089840: set-wise difference of A153829 and A153830.

Original entry on oeis.org

68, 73, 74, 83, 84, 87, 88, 183, 184, 189, 190, 199, 202, 203, 252, 254, 261, 262, 268, 269, 270, 271, 515, 537, 539, 573, 575, 591, 593, 871, 894, 895, 990, 995, 996, 1110, 1132, 1134, 1466, 1489, 1490, 1585, 1590, 1591, 1600, 1601, 1604, 1605, 2213

Offset: 0

Antti Karttunen, Jan 07 2009

nonn

The terms give the positions to bijections in A089840 which preserve A153835/A127302 (the non-oriented form of binary trees), but do not extend uniquely to automorphisms of an infinite binary tree.

A. Karttunen, Table of n, a(n) for n = 0..2933
A. Karttunen, C-program for computing the initial terms of this sequence

Cf. also A153826, A153827, A153828, A153832, A153833.

cp's OEIS Frontend

A153829 Index sequence to A089840: positions of bijections that preserve A153835, or equivalently, A127302 (the non-oriented form of binary trees).

Views

Author

Keywords

Comments

Links

Crossrefs

A154103 Auxiliary array for computing A153835.

Views

Author

Keywords

Comments

Links

Crossrefs

A069770 Signature permutation of the first non-identity, nonrecursive Catalan automorphism in table A089840: swap the top branches of a binary tree. An involution of nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A153831 Index sequence to A089840: set-wise difference of A153829 and A153830.

Views

Author

Keywords

Comments

Links

Crossrefs