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 1-10 of 17 results. Next

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

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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.

Links

Crossrefs

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).

Extensions

Further comments and constructive implementation of Scheme-function (*A089854) 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

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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.

Links

Crossrefs

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).

Extensions

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

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

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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.

Links

Crossrefs

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).

Extensions

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

A089856 Involution of natural numbers induced by Catalan Automorphism *A089856 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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....-->....x...A...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((c . b) . a) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism swaps caar 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.

Links

Crossrefs

Row 10 of A089840. a(n) = A073269(A069770(n)) = A069770(A073270(n)) = A057163(A089852(A057163(n))).
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).

Extensions

Further comments and constructive implementation of Scheme-function (*A089856) added by Antti Karttunen, Jun 04 2011

A085159 Permutation of natural numbers induced by the Catalan bijection gma085159 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085160. a(n) = A085161(A085160(A085161(n))) = A085169(A082315(A085170(n))) = A074684(A082315(A074683(n))) = A085173(A085173(n)). Occurs in A073200. Cf. also A085165-A085168, A086429. Scheme-function app-to-xrt given in A085203.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085160 Permutation of natural numbers induced by the Catalan bijection gma085160 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 18, 11, 20, 12, 14, 21, 13, 15, 19, 22, 23, 45, 24, 44, 46, 25, 48, 26, 42, 49, 27, 43, 47, 50, 28, 54, 29, 53, 55, 30, 57, 31, 37, 58, 32, 38, 56, 59, 33, 61, 34, 39, 62, 35, 40, 51, 63, 36, 41, 52, 60, 64, 65, 129, 66, 128, 130
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085159. a(n) = A085161(A085159(A085161(n))) = A085169(A082316(A085170(n))) = A074684(A082316(A074683(n))) = A085174(A085174(n)). Occurs in A073200. Cf. also A085165-A085168, A086430.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A086430 Permutation of natural numbers induced by the Catalan bijection gma086430 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086429. a(n) = A086428(A086428(n)) = A086431(A086429(A086431(n))) = A057164(A085160(A057164(n))) = A086425(A082316(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A089864 Involution of natural numbers induced by the Catalan automorphism gma089864 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 12, 13, 11, 9, 10, 15, 14, 19, 21, 22, 16, 20, 17, 18, 31, 32, 34, 35, 36, 30, 33, 28, 23, 24, 29, 25, 26, 27, 40, 41, 39, 37, 38, 52, 51, 56, 58, 59, 60, 62, 63, 64, 43, 42, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 87, 88, 90, 91, 92, 96, 97, 99
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This "Catalan bijection" 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.....B..A.D..C.......B...C.......C...B.......A...B........B...A...
..\./.\./.......\./.\./.........\./.........\./.........\./..........\./....
...x...x....-->..x...x.......()..x..-->..()..x...........x..()...-->..x..().
....\./...........\./.........\./.........\./.............\./..........\./..
.....x.............x...........x...........x...............x............x...
i.e. we apply A069770 (that is, the corresponding automorphism) both to the left and right subtree of a binary tree and fix both the empty tree and the tree of one internal node.

Examples

			To obtain this 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 transforms to, as follows:
...................one tree of one internal........2 trees of 2 internal nodes
..empty tree.........(non-leaf) node.................................
........................................................\/.......\/..
......x......................\/........................\/.........\/.
n=....0......................1..........................2..........3.
a(n)=.0......................1..........................2..........3.(all these trees are fixed by this transformation)
however, the next 5 trees, with 3 internal nodes, in range [A014137[2], A014138[2]] = [4,8] change as follows:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
....................|.........................
....................|.........................
....................V.........................
......\/.........\/.............\/.........\/.
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
a(n)=..5........4........6........8........7..
thus we obtain the first nine terms of this sequence: 0,1,2,3,5,4,6,8,7,...

Links

Crossrefs

a(n) = A089859(A089859(n)) = A089863(A089863(n)). Row 1654694 of A089840.
Number of cycles: A089402. Number of fixed points: A089408. Max. cycle size & LCM of all cycle sizes: A046698 (in range [A014137(n-1)..A014138(n-1)] of this permutation).

A086429 Permutation of natural numbers induced by the Catalan bijection gma086429 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 21, 11, 18, 12, 14, 15, 13, 19, 20, 22, 23, 45, 24, 44, 58, 25, 46, 26, 42, 43, 27, 56, 57, 63, 28, 48, 29, 47, 59, 30, 49, 31, 37, 38, 32, 39, 40, 41, 33, 50, 34, 51, 52, 35, 53, 54, 55, 36, 60, 61, 62, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086430. a(n) = A086427(A086427(n)) = A086431(A086430(A086431(n))) = A057164(A085159(A057164(n))) = A086425(A082315(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082313 Involution of natural numbers: A057501-conjugate of A057164.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 6, 7, 5, 9, 21, 14, 18, 13, 11, 22, 16, 17, 12, 19, 20, 10, 15, 23, 58, 37, 49, 35, 28, 62, 42, 46, 32, 51, 55, 27, 41, 25, 59, 39, 50, 36, 30, 63, 44, 45, 31, 53, 54, 26, 40, 33, 64, 47, 48, 34, 56, 57, 24, 38, 60, 61, 29, 43, 52, 65, 170, 107, 142, 100
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003. Proposed by Wouter Meeussen in Dec 15 2001

Keywords

Comments

Note: This is isomorphic with Meeussen's "skewcatacycleft" operation acting on the interpretation (gg) of the exercise 19 by Stanley.

Links

Crossrefs

a(n) = A069888(A057502(n)). Occurs in A073200 as row 604463486276865131809167. Cf. also A082314, A082315, A082333, A082334.
Number of cycles: A007123. Number of fixed-points: A001405. Max. cycle size: A046698. LCM of cycle sizes: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057501(A057164(A057502(n)))
Showing 1-10 of 17 results. Next

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