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 1-8 of 8 results.

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

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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.

A090828 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A085163/A085164.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 37, 95, 285, 876, 2862, 9582, 33022, 115454, 409721, 1467781, 5305211, 19302774, 70662612, 259980148, 960908596
Offset: 0

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Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Links

A090829 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A085163/A085164.

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 12, 14, 20, 20, 28, 28, 36, 36, 44, 44, 52, 52, 60, 60, 68
Offset: 0

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Links

Crossrefs

Compare the bisections to A017113, A086570.

A090830 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A085163/A085164.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 120, 840, 1680, 5040, 5040, 55440, 110880, 1441440, 1441440, 1441440, 1441440, 24504480, 24504480, 465585120, 931170240
Offset: 0

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Author

Antti Karttunen, Dec 20 2003

Keywords

Links

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

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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).

A085171 Permutation of natural numbers induced by the Catalan bijection gma085171 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, 15, 21, 18, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 53, 51, 56, 47, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 29, 40, 38, 43, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085172. Cf. also A085163, A085164, A085203.

A085172 Inverse permutation to A085171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 21, 11, 18, 12, 14, 20, 13, 15, 19, 22, 23, 45, 24, 44, 58, 25, 46, 26, 42, 57, 27, 43, 56, 63, 28, 48, 29, 47, 59, 30, 49, 31, 37, 54, 35, 38, 53, 62, 33, 50, 32, 39, 55, 34, 40, 51, 61, 36, 41, 52, 60, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085171. Cf. also A085163, A085164, A085203.

A123719 An involution of nonnegative integers: signature permutation of Catalan automorphism which is obtained with recursion schema RIBS from automorphism *A085161.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

Recursion schema RIBS is defined in A122200. Number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation is given by INVERT transform of A001405, appropriately shifted.

Links

Crossrefs

a(n) = A085160(A085163(n)). A085163(n) = A085159(a(n)).
Showing 1-8 of 8 results.

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