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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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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

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Author

Antti Karttunen, Jun 23 2003

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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.

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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).

A086434 Contraction-permutation of A085170: a(n) = A082853(A085170(A081291(n))).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 6, 10, 12, 11, 7, 13, 9, 8, 14, 15, 16, 18, 17, 28, 29, 33, 35, 34, 30, 36, 32, 31, 19, 20, 37, 39, 38, 24, 41, 26, 25, 21, 40, 27, 23, 22, 42, 43, 44, 46, 45, 47, 48, 52, 54, 53, 49, 55, 51, 50, 84, 85, 86, 88, 87, 98, 99, 103, 105, 104, 100, 106, 102, 101
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A086433.
Showing 1-2 of 2 results.

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