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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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).

A086431 Involution of natural numbers induced by the Catalan bijection gma086431 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection reflects the interpretations (pp)-(rr) of Stanley, obtained with the "descending slope mapping" from the Dyck paths encoded by A014486.

Examples

			Map the Dyck paths (Stanley's interpretation (i)) to noncrossing Murasaki-diagrams (Stanley's interpretation (rr)) by drawing a vertical line above each descending 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 the Catalan bijection gma086431 gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
.....___________..................................
....|...._..|...|.................................
....|...|.|||..||..................___________....
....|...|.|||..||.................|.._....|...|...
....|../\/\||..||..i.e..equal.to..|.|.|.|.|.|.|...
....|./....\|./\|.................|.|.|.|.|.|.|...
.../\/......\/..\.................|.|.|.|.|.|.|...
...10111010001100=11916=A014486(296)
So we have A086431(250)=296 and A086431(296)=250.

Links

Crossrefs

a(n) = A057164(A085161(A057164(n))) = A086425(A057164(A086426(n))). Occurs in A073200. Cf. also A086427, A086430.
Number of cycles: A007123. Number of fixed points: A001405. (In range [A014137(n-1)..A014138(n-1)] of this permutation.).

A120705 Permutation of natural numbers induced by the Catalan bijection gma120705 acting on the binary trees encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

When the automorphisms A120705/A120705 act on the full Stern-Brocot tree (A007305/A047679), which is an infinite binary tree, they will move each fraction r to the position of 2*r (or r/2 respectively). See comments at A065249 and A065251. (Proof in preparation, to be published.)

Links

Crossrefs

Inverse of A120706. Cf. A074680.
Number of cycles: A120707. Number of fixed-points: A019590. Max. cycle size: A120708. LCM of cycle sizes: A120709.

A120706 Permutation of natural numbers induced by the Catalan bijection gma120706 acting on the binary trees encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 5, 4, 15, 14, 16, 18, 17, 19, 20, 22, 11, 12, 21, 13, 10, 9, 39, 40, 41, 38, 37, 43, 42, 47, 50, 49, 44, 45, 46, 48, 52, 51, 53, 55, 54, 60, 61, 62, 29, 28, 63, 30, 32, 31, 56, 57, 64, 33, 34, 59, 36, 25, 26, 58, 35, 27, 24, 23, 113, 112, 114, 116, 115
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

See comments at A120705.

Links

Crossrefs

Inverse of A120705. Cf. A074679.
Number of cycles: A120707. Number of fixed-points: A019590. Max. cycle size: A120708. LCM of cycle sizes: A120709.

A082339 Permutation of natural numbers induced by the Catalan bijection gma082339 acting on the parenthesizations encoded by A014486/A063171.

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, 19, 21, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 34, 33, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 40, 39, 41, 54, 55, 53, 51, 52, 57, 56, 58, 59, 61, 60, 62, 63, 64, 65, 66, 68, 67, 69
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082340. Occurs in A073200 as row 1796. Cf. also A072797, A082337-A082341.
Max. cycle size and LCMs of cycle sizes: A016116 (to be checked). (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082351 Permutation of natural numbers induced by the Catalan bijection gma082351 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082352. a(n) = A057163(A082353(A057163(n))). Cf. also A074679-A074680, A082355-A082356, A082361-A082362.

A082352 Permutation of natural numbers induced by the Catalan bijection gma082352 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082351. a(n) = A057163(A082354(A057163(n))). Cf. also A074679-A074680, A082355-A082356, A082361-A082362.

A073191 Number of separate orbits/cycles to which the Catalan bijections A072796/A072797 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 96, 305, 1007, 3389, 11636, 40498, 142714, 507870, 1823040, 6591885, 23989419, 87795473, 322922652, 1193058230, 4425547638, 16475756738, 61539293424, 230548633954, 866095934598, 3261868457698, 12313423931624
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Crossrefs

Occurs for first time in A073201 as row 1.

Formula

a(n) = (A000108(n)+A073190(n))/2.

A073193 Number of separate orbits/cycles to which the Catalan bijection A057508 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 93, 292, 965, 3238, 11126, 38708, 136486, 485820, 1744677, 6310584, 22973793, 84103302, 309429066, 1143487428, 4242631626, 15798011604, 59018856522, 221143860936, 830895360978, 3129747395548, 11816242209260
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Crossrefs

Occurs for first time in A073201 as row 168.

Formula

a(n) = (A000108(n)+A073192(n))/2

A082345 Permutation of natural numbers induced by the Catalan bijection gma082345 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082346. Occurs in A073200 as row 66. Cf. also A069767, A073288-A073289, A082347-A082348.
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