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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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

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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).

A085165 A057163-conjugate of A085159.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 12, 16, 17, 21, 10, 18, 11, 14, 15, 13, 19, 20, 22, 23, 26, 30, 31, 35, 42, 43, 44, 45, 49, 56, 57, 58, 63, 24, 32, 47, 46, 59, 25, 48, 28, 37, 38, 29, 39, 40, 41, 27, 50, 33, 51, 52, 34, 53, 54, 55, 36, 60, 61, 62, 64, 65, 68, 72, 73, 77
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085166. a(n) = A057163(A085159(A057163(n))) = A085162(A085166(A085162(n))). Occurs in A073200. Cf. also A085162, A086429, 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).

A085166 A057163-conjugate of A085160.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 10, 19, 17, 18, 11, 12, 15, 20, 21, 13, 22, 23, 37, 42, 24, 51, 44, 47, 25, 26, 38, 53, 56, 27, 60, 45, 46, 48, 49, 50, 28, 29, 30, 31, 40, 39, 43, 32, 52, 54, 55, 57, 58, 59, 33, 34, 35, 41, 61, 62, 63, 36, 64, 65, 107, 121, 66, 149
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085165. a(n) = A057163(A085160(A057163(n))) = A085162(A085165(A085162(n))). Occurs in A073200. Cf. also A085162, A086429, 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).
Showing 1-3 of 3 results.

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