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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A085168 Inverse permutation to A085167.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 8, 6, 5, 17, 9, 18, 14, 10, 20, 11, 21, 16, 12, 22, 19, 15, 13, 45, 23, 46, 37, 24, 48, 25, 49, 42, 26, 50, 51, 38, 27, 54, 28, 55, 39, 29, 57, 30, 58, 44, 31, 59, 47, 40, 32, 61, 33, 62, 53, 34, 63, 56, 43, 35, 64, 60, 52, 41, 36, 129, 65, 130, 107, 66
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

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Inverse: A085167. a(n) = A069770(A085160(n)). Occurs in A073200. Cf. also A074679, A074680, A085203.

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

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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.

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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).

A073281 Self-inverse permutation of natural numbers induced by the composition of the Catalan bijections A072796 and A073269.

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 8, 4, 6, 17, 18, 12, 11, 15, 20, 13, 21, 9, 10, 22, 14, 16, 19, 45, 46, 48, 49, 50, 31, 32, 30, 28, 29, 40, 39, 43, 52, 54, 55, 34, 33, 41, 57, 35, 58, 23, 24, 59, 25, 26, 27, 61, 36, 62, 37, 38, 63, 42, 44, 47, 64, 51, 53, 56, 60, 129, 130, 132, 133, 134
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

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Occurs for first time in A073200 as row 15. This is a conjugate of A069770, so the fixed element and the cycle counts are same as for A069770: "Aerated Catalan numbers" and A007595.

Formula

a(n) = A072796(A073269(n)) = A072796(A069770(A072796(n))).

A073294 Permutation A069767 applied four times or permutation A073290 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 13, 12, 14, 15, 16, 18, 17, 19, 20, 22, 21, 26, 27, 25, 24, 23, 29, 28, 33, 35, 36, 30, 34, 32, 31, 38, 37, 39, 41, 40, 42, 43, 47, 49, 50, 44, 48, 46, 45, 51, 52, 53, 55, 54, 60, 61, 63, 64, 56, 57, 62, 59, 58, 73, 74, 76, 77, 78, 72, 75
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

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Inverse permutation: A073295. Cf. A073290-A073299. Occurs in A073200 as row 142647371.

Formula

a(n) = A069767(A069767(A069767(A069767(n)))) = A073290(A073290(n)).

A073296 Permutation A069767 applied five times or composition of the permutations A073290 & A073292.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 18, 17, 20, 22, 21, 16, 19, 15, 13, 12, 14, 11, 9, 10, 49, 50, 48, 46, 45, 55, 54, 61, 63, 64, 57, 62, 59, 58, 47, 44, 53, 60, 56, 43, 52, 41, 35, 36, 40, 34, 32, 31, 42, 51, 39, 33, 30, 37, 28, 24, 23, 38, 29, 25, 27, 26, 142, 143, 146, 147, 148
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

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Crossrefs

Inverse permutation: A073297. Cf. A073290-A073299. Occurs in A073200 as row 36169536789029067.

Formula

a(n) = A069767^5(n) = A073290(A073292(n)).

A073298 Permutation A069767 applied six times or permutation A073290 "cubed" or permutation A073292 "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, 35, 36, 34, 32, 31, 33, 30, 28, 24, 23, 29, 25, 27, 26, 41, 40, 39, 37, 38, 52, 51, 60, 63, 64, 56, 62, 59, 58, 43, 42, 53, 61, 57, 44, 54, 46, 45, 47, 55, 48, 50, 49, 100, 101, 104, 105, 106
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073299. Cf. A073290-A073297. Occurs in A073200 as row 2596188043348680205834878944583883.

Formula

a(n) = A069767^6(n) = A073290(A073290(A073290(n))) = A073292(A073292(n))

A082314 Involution of natural numbers: A057502-conjugate of A057164.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 11, 10, 12, 13, 21, 22, 20, 17, 18, 19, 16, 14, 15, 23, 28, 25, 30, 33, 24, 29, 26, 31, 34, 27, 32, 35, 36, 58, 62, 59, 63, 64, 57, 61, 54, 45, 48, 55, 46, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 39, 52, 43, 38, 40, 41, 65, 79, 70, 84, 93
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

a(n) = A057502(A069889(n)). Occurs in A073200 as row 2361759710983228099211. Cf. also A082313.
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) = A057502(A057164(A057501(n)))

A082322 Permutation A057502 applied five times ("^5").

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 4, 6, 17, 20, 12, 10, 15, 16, 19, 11, 9, 14, 22, 21, 18, 13, 45, 54, 31, 26, 40, 44, 53, 30, 25, 39, 63, 59, 50, 36, 46, 55, 32, 27, 41, 42, 51, 28, 23, 37, 62, 58, 49, 35, 43, 52, 29, 24, 38, 61, 57, 48, 34, 60, 56, 47, 33, 64, 129, 157, 87, 73, 115
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

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Crossrefs

Inverse of A082321. Occurs in A073200. Cf. also A082323-A082324.

Formula

a(n) = A082316(A082318(n))

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).

A082320 Permutation A057502 applied four times ("^4"), permutation A082316 squared.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082319. Occurs in A073200. Cf. also A082321-A082324.

Formula

a(n) = A082316(A082316(n))
Previous Showing 81-90 of 91 results. Next

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