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 13 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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

A086433 Contraction-permutation of A085169: a(n) = A082853(A085169(A081291(n))).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 6, 10, 13, 12, 7, 9, 8, 11, 14, 15, 16, 18, 17, 28, 29, 37, 41, 40, 33, 36, 35, 39, 19, 20, 24, 27, 26, 21, 23, 22, 25, 30, 32, 31, 38, 34, 42, 43, 44, 46, 45, 47, 48, 52, 55, 54, 49, 51, 50, 53, 84, 85, 86, 88, 87, 112, 113, 126, 131, 130, 121, 125, 124, 129
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A086434.

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Crossrefs

Cf. A014137, A014138, A085169, A085170.

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

Views

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).

A085170 Permutation of natural numbers induced by the Catalan bijection gma085170 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, 21, 20, 16, 22, 18, 17, 23, 24, 25, 27, 26, 28, 29, 33, 35, 34, 30, 36, 32, 31, 37, 38, 39, 41, 40, 51, 52, 56, 58, 57, 53, 59, 55, 54, 42, 43, 60, 62, 61, 47, 64, 49, 48, 44, 63, 50, 46, 45, 65, 66, 67, 69, 68
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085169 (see comments there). a(n) = A086434(A082853(n))+A082852(n). Cf. also A074683, 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

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: 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

Views

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

A089867 Permutation of natural numbers induced by the Catalan bijection gma089867 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, 64, 63, 56, 57, 59, 58, 62, 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 A085169 to the left subtree and keep the right subtree intact.

Links

Crossrefs

Inverse of A089868.
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).
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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