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 11 results. Next

A123719 An involution of nonnegative integers: signature permutation of Catalan automorphism which is obtained with recursion schema RIBS from automorphism *A085161.

Original entry on oeis.org

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

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Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

Recursion schema RIBS is defined in A122200. Number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation is given by INVERT transform of A001405, appropriately shifted.

Links

Crossrefs

a(n) = A085160(A085163(n)). A085163(n) = A085159(a(n)).

A085169 Permutation of natural numbers induced by the Catalan bijection gma085169 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, 22, 21, 16, 18, 17, 20, 23, 24, 25, 27, 26, 28, 29, 33, 36, 35, 30, 32, 31, 34, 37, 38, 39, 41, 40, 51, 52, 60, 64, 63, 56, 59, 58, 62, 42, 43, 47, 50, 49, 44, 46, 45, 48, 53, 55, 54, 61, 57, 65, 66, 67, 69, 68
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

A parenthesization is fixed by the Catalan bijections A085169/A085170 if and only if no other elements than () and (()) occur at its top-level: (); ()(),(()); ()()(),()(()),(())(); ()()()(),()()(()),()(())(),(())()(),(())(()); ... There is a simple bijection between these and Zeckendorf-expansions, explaining why Fibonacci numbers gives the number of fixed points of this permutation.
In addition to "rising slope" and "descending slope" mappings from Dyck paths to noncrossing Murasaki-diagrams as illustrated in A085161 and A086431 there is also a mapping where we insert a vertical stick after every second parenthesis and connect those that are on the same level without any intermediate points below. This Catalan bijection converts between these two mappings. See the illustration at example lines.

Examples

			.........................
..._____....________.....
..|.....|..|.....|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..|..|..|..|..|..|..|....
..1((2))3((4((5))6()7))..
...(())(((())()))........
...11001111001000=13256=A014486(368)
To obtain the same Murasaki diagram using the "rising slope mapping" illustrated in A085161, we should use the following Dyck path, encoded by 360th binary string in A014486/A063171:
....___.._____...........
...|...||...|.|..........
...||..|||..|.|..........
...||..|||..|.|..........
...||..||/\.|.|..........
...|/\.|/..\/\/\.........
.../..\/........\........
...11001110010100=13204=A014486(360)
So we have A085169(368)=360 and A085170(360)=368.

Links

Crossrefs

Inverse: A085170. a(n) = A086433(A082853(n))+A082852(n). A074684 = A083925(A085169(A057548(n))). Cf. also 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).

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.).

A085163 Permutation of natural numbers induced by the Catalan bijection gma085163 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, 18, 21, 15, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 56, 51, 53, 47, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 46, 49, 38, 32, 59, 58, 62, 40, 29, 55, 35, 43, 27, 50, 63, 52, 41, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085164. a(n) = A085161(A057508(n)). Occurs in A073200. Cf. also A085171, A085172. Scheme-function app-to-xrt given in A085203.
Number of fixed points 1, 1, 2, 2, 3, 4, 7, 11, 21, 36, 71, ... in range [A014137(n-1)..A014138(n-1)] of this permutation appears to be A051920 after the initial two ones.

A085164 Inverse permutation to A085163.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085163. a(n) = A057508(A085161(n)). Occurs in A073200. Cf. also A085171, A085172.
Number of fixed points: A051920. See comment at A085163.

A085162 Permutation of natural numbers induced by Catalan Automorphism *A085162.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 17, 16, 12, 21, 14, 15, 11, 10, 18, 20, 19, 13, 22, 23, 45, 44, 31, 58, 42, 43, 30, 26, 49, 57, 56, 35, 63, 37, 40, 39, 38, 41, 28, 29, 25, 24, 46, 48, 47, 32, 59, 54, 55, 53, 51, 52, 34, 33, 27, 50, 62, 61, 60, 36, 64, 65, 129, 128, 87, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

a(n) = A057163(A085161(A057163(n))). Cf. also A085165, A085166, A086431. Occurs in A073200.
Number of cycles: A007123. Number of fixed points: A001405 (in each range limited by A014137 and A014138).

A085173 Permutation of natural numbers induced by the Catalan bijection gma085173 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

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

Links

Crossrefs

Inverse: A085174. a(n) = A085161(A085174(A085161(n))) = A085169(A057501(A085170(n))) = A074684(A057501(A074683(n))). Occurs in A073200. Cf. also A085159 (whole step rotate), A086427.
Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085174 Permutation of natural numbers induced by the Catalan bijection gma085174 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 21, 15, 14, 18, 16, 20, 13, 11, 17, 12, 10, 9, 64, 60, 63, 52, 51, 59, 56, 62, 41, 39, 58, 40, 38, 37, 50, 47, 49, 43, 42, 55, 53, 61, 36, 33, 54, 35, 29, 28, 46, 44, 57, 32, 30, 48, 34, 27, 25, 45, 31, 26, 24, 23, 196, 191, 195, 178, 177
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

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

Links

Crossrefs

Inverse: A085173. a(n) = A085161(A085173(A085161(n))) = A085169(A057502(A085170(n))) = A074684(A057502(A074683(n))). Occurs in A073200. Cf. also A085160 (whole step rotate), A086428.
Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).
Showing 1-10 of 11 results. Next

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