A089840 -id:A089840 - cp's OEIS frontend

A073269 Permutation of natural numbers induced by Catalan Automorphism *A073269 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.....B...C.......A...C
......\./.........\./
...A...x...-->... .x...B...............A..().........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
(a . (b . c)) --> ((a . c) . b) ___ (a . ()) --> (() . a)
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Row 14 of A089840. Inverse permutation: A073270. a(n) = A069770(A072796(n)).

Further comments, a mail-address and Scheme-implementation of this automorphism added by Antti Karttunen, Jun 04 2011

A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 16, 20, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...........B...A
..\./.............\./
...x...C....-->....x...C...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . a) . c) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.

Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

Further comments and constructive implementation of Scheme-function (*A089854) added by Antti Karttunen, Jun 04 2011

A122290 Signature permutations of KROF-transformations of Catalan automorphisms in table A122202.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 18, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 17, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 10, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 22, 14, 13, 15

Offset: 0

Antti Karttunen, Sep 01 2006

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122202 with the recursion scheme "KROF", or equivalently row n is obtained as KROF(KROF(n-th row of A089840)). See A122202 for the description of KROF. Each row occurs only once in this table. Inverses of these permutations can be found in table A122289.

A. Karttunen, paper in preparation, draft available by e-mail.

Index entries for signature-permutations of Catalan automorphisms

The known rows of this table: row 0 (identity permutation): A001477, row 1: A122351, row 2: A122364. See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288.

A089853 Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 7, 8, 16, 19, 14, 9, 10, 15, 11, 12, 17, 18, 13, 20, 21, 22, 44, 47, 53, 56, 60, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 45, 46, 32, 48, 49, 50, 41, 33, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 128, 131, 137, 140, 144

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
...B...C...........A...B
....\./.............\./
.A...x....-->....C...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (c . (a . b)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism rotates car, cadr and cddr of an S-exp if its length > 1.
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A089851. a(n) = A072796(A089850(n)) = A057163(A089855(A057163(n))). Row 6 of A089840.

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Further comments and constructive implementation of Scheme-function (*A089853) added by Antti Karttunen, Jun 04 2011

A073270 Permutation of natural numbers induced by Catalan Automorphism *A073270 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...............C...B
..\./.................\./
...x...C...-->.....A...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (a . (c . b)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Row 19 of A089840. Inverse permutation: A073269. a(n) = A072796(A069770(n)).

A graphical description and Scheme-implementations of automorphism added by Antti Karttunen, Jun 04 2011

A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 12, 13, 11, 9, 10, 15, 14, 16, 17, 18, 19, 20, 21, 22, 31, 32, 34, 35, 36, 30, 33, 28, 23, 24, 29, 25, 26, 27, 40, 41, 39, 37, 38, 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, 99

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C...........C...B
....\./.............\./
.A...x....-->....A...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (a . (c . b)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism swaps cadr and cddr of an S-exp if its length > 1.
Look at the example in A069770 to see how this will produce the given sequence of integers.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A069770(A089859(n)) = A089863(A069770(n)) = A057163(A089854(A057163(n))). Row 3 of A089840. Row 3771 of A122203 and row 3677 of A122204.

Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

The new mail-address, a graphical explanation and constructive implementation of Scheme-function (*A089850) added by Antti Karttunen, Jun 04 2011

A122289 Signature permutations of FORK-transformations of Catalan automorphisms in table A122201.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 18, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 17, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 10, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 22, 14, 13, 12

Offset: 0

Antti Karttunen, Sep 01 2006

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122201 with the recursion scheme "FORK", or equivalently row n is obtained as FORK(FORK(n-th row of A089840)). See A122201 for the description of FORK. Each row occurs only once in this table. Inverses of these permutations can be found in table A122290.

A. Karttunen, paper in preparation, draft available by e-mail.

Index entries for signature-permutations of Catalan automorphisms

The known rows of this table: row 0 (identity permutation): A001477, row 1: A122351, row 2: A122363. See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288.

A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 8, 16, 19, 15, 12, 13, 14, 11, 9, 17, 18, 10, 20, 21, 22, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 37, 28, 23, 45, 46, 24, 48, 49, 50, 38, 29, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 128, 131, 137, 140, 144

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
...B...C...........B...A
....\./.............\./
.A...x....-->....C...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (c . (b . a)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism swaps car and cddr of an S-exp if its length > 1, if possible, otherwise keeps it intact.
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A069770(A089858(n)) = A089861(A069770(n)) = A057163(A089856(A057163(n))). Row 5 of A089840.

Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

Further comments and constructive implementation of Scheme-function (*A089852) added by Antti Karttunen, Jun 04 2011

A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 11, 12, 13, 17, 18, 20, 21, 22, 16, 19, 14, 15, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 42, 51, 37, 38, 43, 52, 39, 40, 41, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...........B...C
..\./.............\./
...x...C....-->....x...A...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . c) . a) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp if possible, i.e., if car-side is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A089857. a(n) = A089860(A069770(n)) = A069770(A074680(n)) = A057163(A089853(A057163(n))). Row 9 of A089840.

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Further comments and constructive implementation of Scheme-function (*A089855) added by Antti Karttunen, Jun 04 2011

A089857 Permutation of natural numbers induced by Catalan Automorphism *A089857 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 6, 7, 9, 10, 11, 12, 13, 21, 22, 19, 14, 15, 20, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 58, 59, 62, 63, 64, 56, 60, 51, 37, 38, 52, 39, 40, 41, 57, 61, 53, 42, 43, 54, 44, 45, 46, 55, 47, 48, 49, 50, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

.A...B...........C...A
..\./.............\./
...x...C....-->....x...B...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((c . a) . b) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp, i.e., if car-side is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

Row 11 of A089840. Inverse of A089855. a(n) = A074679(A069770(n)) = A069770(A089862(n)) = A057163(A089851(A057163(n))).

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Further comments and constructive implementation of Scheme-function (*A089857) added by Antti Karttunen, Jun 04 2011

cp's OEIS Frontend

A073269 Permutation of natural numbers induced by Catalan Automorphism *A073269 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A122290 Signature permutations of KROF-transformations of Catalan automorphisms in table A122202.

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A089853 Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A073270 Permutation of natural numbers induced by Catalan Automorphism *A073270 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A122289 Signature permutations of FORK-transformations of Catalan automorphisms in table A122201.

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A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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A089857 Permutation of natural numbers induced by Catalan Automorphism *A089857 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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