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 12 results. Next

A086587 Least common multiple of cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutations A085169/A085170.

Original entry on oeis.org

1, 1, 1, 2, 10, 90, 1260, 167580, 10345048560, 210224307704851440, 142378995493242911206243440, 4409130655192711420325660927780160, 308972448405145190275995459920449062174478109373358971999360
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

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

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Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A086433.

A086585 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutations A085169/A085170.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 40, 86, 198, 438, 982, 2218, 5072, 11642
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

Crossrefs

Cf. A014137, A014138, A085169, A085170.

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

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

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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).

A086586 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutations A074681/A074682 & A074683/A074684.

Original entry on oeis.org

1, 1, 2, 5, 9, 28, 57, 253, 842, 3753, 10927, 15014, 130831, 218961, 967104, 3767216, 29715310, 89923607, 314897868, 785059994
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

Shifted once right (beginning as 1,1,1,2,5,9,...) this is maximum cycle size (in the same range) of permutations A085169/A085170, shifted twice right (beginning as 1,1,1,1,2,5,9,...) this is the maximum cycle size in permutations A089867/A089868 and A089869/A089870.

Links

A089868 Permutation of natural numbers induced by the Catalan bijection gma089868 acting on the parenthesizations/binary trees encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

This Catalan bijection arises when we apply the Catalan bijection A085170 to the left subtree and keep the right subtree intact.

Links

Crossrefs

Inverse of A089867.
Number of cycles: A089846. Number of fixed-points: A090826. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

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).

A089870 Permutation of natural numbers induced by the Catalan bijection gma089870 acting on the parenthesizations/binary trees encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

This Catalan bijection arises when we apply the Catalan bijection A085170 to each top-level subtree (sub-parenthesization).

Links

Crossrefs

Inverse of A089869.
Number of cycles: A090827. Number of fixed-points: A000129. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).
Showing 1-10 of 12 results. Next

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