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.

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A089861 Permutation of natural numbers induced by Catalan Automorphism *A089861 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 6, 5, 17, 18, 20, 21, 22, 9, 10, 14, 16, 19, 11, 15, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 32, 33, 41, 34, 35, 36, 129, 130, 132, 133, 134, 138
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...............A...C
..\./.................\./
...x...C...-->.....B...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (b . (a . c)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 18 of A089840. Inverse of A089858. a(n) = A089852(A069770(n)) = A069770(A072797(n)) = A057163(A073269(A057163(n))).
Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A089862 Permutation of natural numbers induced by Catalan automorphism *A089862 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 6, 4, 17, 18, 20, 21, 22, 12, 13, 15, 16, 19, 11, 14, 9, 10, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 24, 29, 38, 25, 26, 27, 129, 130, 132, 133, 134
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...A
..\./.................\./
...x...C...-->.....B...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (b . (c . a)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 20 of A089840. Inverse of A089860. a(n) = A089853(A069770(n)) = A069770(A089857(n)) = A069770(A074679(A069770(n))) = A057163(A089860(A057163(n))). Number of cycles: A001683 (seems to be, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A074686 Permutation of natural numbers induced by the Catalan bijection gmA074686! acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 22, 21, 17, 18, 20, 10, 9, 11, 12, 13, 14, 15, 16, 19, 64, 63, 58, 59, 62, 46, 45, 48, 49, 50, 54, 55, 57, 61, 27, 26, 23, 24, 25, 29, 28, 30, 31, 32, 33, 34, 35, 36, 38, 37, 39, 40, 41, 42, 43, 44, 47, 51, 52, 53, 56, 60, 196, 195, 189, 190, 194
Offset: 0

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Links

Crossrefs

Inverse of A074685. a(n) = A057163(A074690(A057163(n))). Occurs in A073200.

A085161 Involution of natural numbers induced by Catalan Automorphism *A085161 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 17, 14, 12, 21, 11, 20, 16, 10, 18, 19, 15, 13, 22, 23, 45, 37, 31, 58, 28, 54, 42, 26, 49, 51, 40, 35, 63, 25, 48, 39, 34, 62, 30, 57, 44, 24, 46, 56, 38, 32, 59, 33, 61, 53, 29, 55, 47, 43, 27, 50, 60, 52, 41, 36, 64, 65, 129, 107, 87, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This automorphism reflects the interpretations (pp)-(rr) of Stanley, obtained from the Dyck paths with the "rising slope mapping" illustrated on the example lines.

Examples

			Map the Dyck paths (Stanley's interpretation (i)) to noncrossing Murasaki-diagrams (Stanley's interpretation (rr)) by drawing a vertical line above each rising slope / and connect those vertical lines that originate from the same height without any lower valleys between, as in illustration below:
..................................................
...._____..___....................................
...|.|...||...|...................................
...|.||..|||..|...................._.___...___....
...|.||..|||..|...................|.|...|.|...|...
...|.||..||/\.|....i.e..equal.to..|.|.|.|.|.|.|...
...|.|/\.|/..\/\..................|.|.|.|.|.|.|...
.../\/..\/......\.................|.|.|.|.|.|.|...
...10110011100100=11492=A014486(250)..............
...()(())((())()).................................
Now this automorphism gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
....___.._____....................................
...|...||...|.|...................................
...||..|||..|.|....................___..._____....
...||..|||..|.|...................|...|.|...|.|...
...||..||/\.|.|....i.e..equal.to..|.|.|.|.|.|.|...
...|/\.|/..\/\/\..................|.|.|.|.|.|.|...
.../..\/........\.................|.|.|.|.|.|.|...
...11001110010100=13204=A014486(360)..............
...(())((())()()).................................
So we have A085161(250)=360 and A085161(360)=250.

Links

Crossrefs

a(n) = A085163(A057508(n)) = A074684(A057164(A074683(n))). Occurs in A073200. Cf. also A085159, A085160, A085162, A085175. Alternative mappings illustrated in A086431 & A085169.
Number of cycles: A007123. Number of fixed points: A001405 (in each range limited by A014137 and A014138).

A085170 Permutation of natural numbers induced by the Catalan bijection gma085170 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085169 (see comments there). a(n) = A086434(A082853(n))+A082852(n). Cf. also A074683, A085159, A085160, A085175.
Number of cycles: A086585. Number of fixed points: A000045. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation).

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

A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Links

Crossrefs

Inverse of A082356. a(n) = A082357(A057163(n)). a(n) = A082363(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.
Differs from A057118 first time at n=42: a(42)=56, while A057118(42)=58.

A082356 Permutation of natural numbers induced by Catalan Automorphism *A082356 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Links

Crossrefs

Inverse of A082355. a(n) = A057163(A082358(n)). a(n) = A082364(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.
Differs from A057117 first time at n=56: a(56)=42, while A057117(56)=44.

A074681 Permutation of natural numbers induced by the Catalan bijection gmA074681! acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 5, 4, 15, 14, 19, 20, 22, 16, 21, 18, 11, 13, 17, 12, 10, 9, 39, 41, 40, 38, 37, 52, 51, 53, 55, 54, 60, 61, 62, 64, 43, 42, 56, 57, 63, 47, 59, 48, 29, 28, 50, 33, 34, 36, 44, 58, 49, 30, 35, 46, 32, 25, 27, 45, 31, 26, 24, 23, 113, 112, 117, 118, 120
Offset: 0

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Links

Crossrefs

Inverse of A074682. a(n) = A057163(A074684(A057163(n))). Cf. A074685, A074687, A074689. Occurs in A073200 as row 5572432.

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).
Previous Showing 21-30 of 142 results. Next

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