A089840 -id:A089840 - cp's OEIS frontend

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

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.

A154122 Signature permutation of a Catalan bijection: row 3747 of A089840.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 12, 13, 9, 10, 11, 15, 14, 16, 17, 18, 19, 20, 21, 22, 31, 32, 34, 35, 36, 23, 24, 25, 26, 27, 28, 29, 30, 33, 40, 41, 37, 38, 39, 43, 42, 44, 45, 46, 47, 48, 49, 50, 52, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 87, 88, 90, 91, 92, 96, 97

Offset: 0

Antti Karttunen, Jan 06 2009

nonn

This bijection of binary trees can be obtained by applying bijection *A074680 to the right hand side subtree and leaving the left hand side subtree intact. See also comments at A154121.
.B...C...............C...D
..\./.................\./
...x...D....-->....B...x.................()..C ........C...()
....\./.............\./...................\./....-->....\./...
.A...x...........A...x.................A...x.........A...x....
..\./.............\./...................\./...........\./.....
...x...............x.....................x.............x......
..............................................................
That is, (a . ((b . c) . d)) -> (a . (b . (c . d)))
or (a . (() . c)) -> (a . (c . ())) if the former is not possible.

Inverse: A154121. a(n) = A069770(A089866(A069770(n))). Cf. A154124.

A089843 Involution of natural numbers which shows the position of the inverse of each non-recursive Catalan automorphism in table A089840.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 8, 11, 10, 9, 17, 18, 19, 21, 20, 12, 13, 14, 16, 15, 22, 23, 26, 25, 24, 27, 28, 35, 44, 43, 36, 33, 40, 29, 32, 37, 42, 39, 34, 41, 38, 31, 30, 45, 46, 49, 48, 47, 50, 51, 58, 67, 66, 59, 56, 63, 52, 55, 60, 65, 62, 57, 64, 61, 54, 53, 68, 69, 72, 71

Offset: 0

Antti Karttunen, Dec 05 2003

nonn

Cf. A089837, A089838.

A123497 Signature permutation of a nonrecursive Catalan automorphism: row 1655089 of table A089840.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 10, 14, 16, 19, 17, 18, 12, 11, 13, 20, 15, 21, 22, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 45, 46, 48, 49, 50, 31, 32, 30, 28, 29, 34, 33, 35, 36, 54, 55, 40, 39, 41, 57, 43, 58, 59, 61, 52, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123498. Row 1655089 of A089840. Used to construct automorphism *A123501. A074680(n) = A083927(a(A057123(n))).

A123498 Signature permutation of a nonrecursive Catalan automorphism: row 1654249 of table A089840.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 10, 17, 16, 18, 11, 20, 12, 14, 15, 13, 19, 21, 22, 23, 24, 25, 26, 27, 45, 46, 44, 42, 43, 48, 47, 49, 50, 28, 29, 54, 53, 55, 30, 57, 31, 37, 38, 32, 39, 40, 41, 33, 61, 34, 51, 52, 35, 56, 58, 59, 36, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123497. Row 1654249 of A089840. Used to construct automorphism *A123502. A074679(n) = A083927(a(A057123(n))).

A129604 Signature-permutation of a Catalan automorphism, row 1654720 of A089840.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 5, 4, 21, 22, 20, 17, 18, 19, 16, 15, 12, 13, 14, 11, 9, 10, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 56, 60, 53, 44, 47, 52, 43, 40, 31, 32, 41, 34, 35, 36, 51, 42, 39, 30, 33, 37, 28, 23, 24, 38, 29, 25, 26, 27, 170, 171, 174, 175, 176

Offset: 0

Antti Karttunen, May 22 2007

nonn

This involution effects the following transformation on the binary trees (labels A,B,C,D refer to arbitrary subtrees located on those nodes and () stands for a terminal node.)
.A..B.C..D.....D..C.B..A.......B...C...C...B........A...B............B...A
..\./.\./.......\./.\./.........\./.....\./..........\./..............\./.
...x...x....-->..x...x.......()..x..-->..x..()........x..()...-->..()..x..
....\./...........\./.........\./.........\./..........\./..........\./...
.....x.............x...........x...........x............x............x....
Note that automorphism *A069770 = FORK(*A129604) = KROF(*A129604). See the definitions given in A122201 and A122202.

a(n) = A069770(A089864(n)) = A089864(A069770(n)). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this involution are given by the same sequences as is the case for example with A069770, A057163 and A122351, that is, A007595 and zero-interspersed A000108.

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.

A154126 Self-inverse signature permutation of a Catalan bijection: row 183 of A089840.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 06 2009

nonn

This bijection of binary trees swaps the left and right subtree of a binary tree, but ONLY if either of them is empty. If both the left and right hand side tree is nonempty, fixes the tree.
.A...B.C...D.......A...B.C...D.....
..\./...\./.........\./...\./........................
...x.....x...--->....x.....x.......A...B.......B...A.
....\.../.............\.../.........\./..--->...\./..
......x.................x............x...........x...
..............................(where either A or B is (), a leaf)
This automorphism demonstrates that not every clause in clause-representations of A089840 is equal to some (minimally represented) element of Thompson's group V.

A. Karttunen, Table of n, a(n) for n = 0..2055
J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathématique, Vol. 42 (1996), pp. 215-256.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A154126. a(n) = A069770(A154125(n)) = A154125(A069770(n)).

A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123495. a(n) = A069770(A082352(n)). Row 65796 of A089840. Used to construct automorphism *A082358. Cf. A069770 and A074680.

A123694 a(n) gives the A089840-index of the nonrecursive Catalan automorphism which is formed from A089840[n] by applying it to the left subtree of a binary tree and leaving the right-hand side subtree intact.

Original entry on oeis.org

0, 7, 91, 92, 93, 94, 95, 114, 115, 116, 117, 118, 4207, 4209, 4211, 4214, 4216, 4299, 4301, 4303, 4305, 4307, 1228, 1229, 1230, 1231, 1232, 1233, 1234, 1235, 1236, 1237, 1238, 1239, 1240, 1241, 1242, 1243, 1244, 1245, 1246, 1247, 1248, 1249, 1250, 1347

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution). See also the comments at A122200.

			When A089840[1] = A069770 (swap binary tree sides) is applied to the left subtree of a binary tree, we get A089840[7] = A089854, thus a(1)=7. When A089840[12] = A074679 is applied to the left subtree of a binary tree, we get A089840[4207] = A089865, thus a(12)=4207.

A. Karttunen, C-program for computing the initial terms of this sequence
A. Karttunen, Prolog-program which illustrates the nonrecursive Catalan automorphisms given on example-lines.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

A154122 Signature permutation of a Catalan bijection: row 3747 of A089840.

Views

Author

Keywords

Comments

Links

Crossrefs

A089843 Involution of natural numbers which shows the position of the inverse of each non-recursive Catalan automorphism in table A089840.

Views

Author

Keywords

Links

Crossrefs

A123497 Signature permutation of a nonrecursive Catalan automorphism: row 1655089 of table A089840.

Views

Author

Keywords

Links

Crossrefs

A123498 Signature permutation of a nonrecursive Catalan automorphism: row 1654249 of table A089840.

Views

Author

Keywords

Links

Crossrefs

A129604 Signature-permutation of a Catalan automorphism, row 1654720 of A089840.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

A154126 Self-inverse signature permutation of a Catalan bijection: row 183 of A089840.

Views

Author

Keywords

Comments

Links

Crossrefs

A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

Views

Author

Keywords

Links

Crossrefs

A123694 a(n) gives the A089840-index of the nonrecursive Catalan automorphism which is formed from A089840[n] by applying it to the left subtree of a binary tree and leaving the right-hand side subtree intact.

Views

Author

Keywords

Comments

Examples

Links