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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A086429 Permutation of natural numbers induced by the Catalan bijection gma086429 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, 21, 11, 18, 12, 14, 15, 13, 19, 20, 22, 23, 45, 24, 44, 58, 25, 46, 26, 42, 43, 27, 56, 57, 63, 28, 48, 29, 47, 59, 30, 49, 31, 37, 38, 32, 39, 40, 41, 33, 50, 34, 51, 52, 35, 53, 54, 55, 36, 60, 61, 62, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086430. a(n) = A086427(A086427(n)) = A086431(A086430(A086431(n))) = A057164(A085159(A057164(n))) = A086425(A082315(A086426(n))). Occurs in A073200.
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).

A073291 Permutation A069768 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 13, 12, 11, 9, 10, 15, 14, 19, 22, 21, 16, 20, 17, 18, 36, 35, 34, 31, 32, 33, 30, 28, 23, 24, 29, 25, 26, 27, 41, 40, 39, 37, 38, 52, 51, 60, 64, 63, 56, 62, 58, 59, 43, 42, 53, 61, 57, 44, 54, 45, 46, 47, 55, 48, 49, 50, 106, 105, 104, 100, 101
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073290. Occurs for first time in A073200 as row 197. Cf. A073292-A073299.

Formula

a(n) = A069768(A069768(n)).

A074682 Permutation of natural numbers induced by the Catalan bijection gmA074682! acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Links

Crossrefs

Inverse of A074681. a(n) = A057163(A074683(A057163(n))). Occurs in A073200.

A082313 Involution of natural numbers: A057501-conjugate of A057164.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 6, 7, 5, 9, 21, 14, 18, 13, 11, 22, 16, 17, 12, 19, 20, 10, 15, 23, 58, 37, 49, 35, 28, 62, 42, 46, 32, 51, 55, 27, 41, 25, 59, 39, 50, 36, 30, 63, 44, 45, 31, 53, 54, 26, 40, 33, 64, 47, 48, 34, 56, 57, 24, 38, 60, 61, 29, 43, 52, 65, 170, 107, 142, 100
Offset: 0

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Author

Antti Karttunen, Apr 17 2003. Proposed by Wouter Meeussen in Dec 15 2001

Keywords

Comments

Note: This is isomorphic with Meeussen's "skewcatacycleft" operation acting on the interpretation (gg) of the exercise 19 by Stanley.

Links

Crossrefs

a(n) = A069888(A057502(n)). Occurs in A073200 as row 604463486276865131809167. Cf. also A082314, A082315, A082333, A082334.
Number of cycles: A007123. Number of fixed-points: A001405. Max. cycle size: A046698. LCM of cycle sizes: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057501(A057164(A057502(n)))

A082317 Permutation A057501 applied thrice ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 4, 8, 17, 20, 12, 10, 15, 16, 19, 11, 9, 14, 22, 21, 18, 13, 45, 48, 54, 57, 61, 31, 34, 26, 24, 29, 40, 38, 43, 52, 44, 47, 53, 56, 60, 30, 33, 25, 23, 28, 39, 37, 42, 51, 63, 64, 59, 58, 62, 50, 49, 46, 55, 36, 35, 32, 27, 41, 129, 132, 138, 141, 145
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082318. Occurs in A073200 as row 11150372599265311570767859136464952601119119. Cf. also A082319-A082324.

Formula

a(n) = A057501(A082315(n))

A082325 Permutation of natural numbers: A057163-conjugate of A057511.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082326. a(n) = A069787(A082326(A069787(n))). a(n) = A082327(A082853(n))+A082852(n). Occurs in A073200 as row 1792. Cf. also A082337-A082338.
Differs from A082342 first time at n=39: a(39)=49, while A082342(39)=48.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057163(A057511(A057163(n)))

A082326 Permutation of natural numbers: A057163-conjugate of A057512.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082325. a(n) = A069787(A082325(A069787(n))). a(n) = A082328(A082853(n))+A082852(n). Occurs in A073200 as row 1794. Cf. also A082337-A082338.
Differs from A082341 first time at n=39: a(39)=49, while A082341(39)=48.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057163(A057512(A057163(n)))

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.

A073197 Permutation of natural numbers induced by the Catalan bijection gmA073197 acting on the parenthesizations as ordered by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Is there an equivalent Catalan bijection in A073200?

Links

Crossrefs

Inverse permutation: A073196. The car/cdr-flipped conjugate of A073208, i.e. A073197(n) = A057163(A073208(A057163(n))). Cf. also A073194-A073199.
The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.

A073207 Permutation of natural numbers induced by the Catalan bijection gmA073207 acting on the parenthesizations as ordered by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 11, 17, 16, 20, 12, 18, 10, 14, 15, 13, 19, 21, 22, 23, 31, 28, 24, 33, 45, 49, 44, 42, 43, 54, 53, 57, 61, 30, 29, 46, 47, 55, 25, 48, 26, 37, 40, 32, 38, 39, 41, 34, 50, 27, 51, 52, 35, 56, 58, 59, 36, 60, 62, 63, 64, 65, 84, 87, 73, 96, 79, 81
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Is there an equivalent Catalan bijection in A073200?

Links

Crossrefs

Inverse permutation: A073208. The car/cdr-flipped conjugate of A073196, i.e. A073207(n) = A057163(A073196(A057163(n))). Cf. also A073205-A073210.
The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
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