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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A073290 Permutation A069767 applied twice ("squared").

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073291. Occurs for first time in A073200 as row 105. A073267 gives essentially (apart from the first two terms) the counts of elements fixed. Cf. A073292-A073299.

Formula

a(n) = A069767(A069767(n)).

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).

A073288 Permutation of natural numbers induced by the Catalan bijection gma073288! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073289. Occurs for first time in A073200 as row 416.
The scheme function gma073286! referred to below given in A073286.

A073289 Permutation of natural numbers induced by the Catalan bijection gma073289! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073288. Occurs for first time in A073200 as row 696.
The scheme function gma073287! referred to below given in A073287.

A082315 Permutation A057501 applied twice ("squared").

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003, Proposed by Wouter Meeussen

Keywords

Links

Crossrefs

Inverse of A082316. a(n) = A082313(A057164(n)). Occurs in A073200 as row 34359740687. Cf. also A082317-A082324.
Number of cycles: A054357. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057501(A057501(n)).

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).

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).

A086431 Involution of natural numbers induced by the Catalan bijection gma086431 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 12, 13, 17, 18, 16, 14, 15, 21, 20, 19, 22, 23, 28, 25, 30, 33, 24, 29, 26, 31, 32, 27, 35, 34, 36, 45, 48, 46, 49, 50, 44, 47, 42, 37, 39, 43, 38, 40, 41, 58, 59, 57, 54, 55, 56, 53, 51, 52, 63, 62, 61, 60, 64, 65, 79, 70, 84, 93
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection reflects the interpretations (pp)-(rr) of Stanley, obtained with the "descending slope mapping" from the Dyck paths encoded by A014486.

Examples

			Map the Dyck paths (Stanley's interpretation (i)) to noncrossing Murasaki-diagrams (Stanley's interpretation (rr)) by drawing a vertical line above each descending 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 the Catalan bijection gma086431 gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
.....___________..................................
....|...._..|...|.................................
....|...|.|||..||..................___________....
....|...|.|||..||.................|.._....|...|...
....|../\/\||..||..i.e..equal.to..|.|.|.|.|.|.|...
....|./....\|./\|.................|.|.|.|.|.|.|...
.../\/......\/..\.................|.|.|.|.|.|.|...
...10111010001100=11916=A014486(296)
So we have A086431(250)=296 and A086431(296)=250.

Links

Crossrefs

a(n) = A057164(A085161(A057164(n))) = A086425(A057164(A086426(n))). Occurs in A073200. Cf. also A086427, A086430.
Number of cycles: A007123. Number of fixed points: A001405. (In range [A014137(n-1)..A014138(n-1)] of this permutation.).
Previous Showing 11-20 of 91 results. Next

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