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 14 results. Next

A089413 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071655/A071656.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 5, 5, 11, 11, 17, 11, 21, 17, 30, 22, 52, 24, 74, 36
Offset: 0

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Author

Antti Karttunen, Nov 29 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.

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Crossrefs

Bisections: A089409, A089416. Cf. A089411.

A089414 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A071655/A071656.

Original entry on oeis.org

1, 1, 2, 3, 12, 29, 97, 378, 775, 2940, 11368, 45671, 113556, 479745, 1094742, 3536813, 15177837, 67116740, 300914463, 519239984, 3251343406
Offset: 0

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Author

Antti Karttunen, Nov 29 2003

Keywords

Links

A089415 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A071655/A071656.

Original entry on oeis.org

1, 1, 2, 6, 12, 870, 13580, 10962, 20425900, 1127689920
Offset: 0

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Author

Antti Karttunen, Nov 29 2003

Keywords

Links

A074680 Signature permutation of the seventeenth nonrecursive Catalan automorphism in table A089840. (Rotate binary tree right if possible, otherwise swap its sides.)

Original entry on oeis.org

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

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Author

Antti Karttunen, Sep 11 2002

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
A...B..............B...C
.\./................\./
..x...C..-->.....A...x................()..B.......B..()
...\./............\./..................\./...-->...\./.
....x..............x....................x...........x..
((a . b) . c) -> (a . (b . c)) __ (() . b) --> (b . ())
That is, we rotate the binary tree right, in case it is possible and otherwise (if the left hand side of a tree is a terminal node) swap the right and left subtree (so that the terminal node ends to the right hand side), i.e. apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.
See also the comments at A074679.

References

  • A. Karttunen, paper in preparation, draft available by e-mail.

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Crossrefs

This automorphism has several variants, where the first clause is same (rotate binary tree to the right, if possible), but something else is done (than just swapping sides), in case the left hand side is empty: A082336, A082350, A123500, A123696. The following automorphisms can be derived recursively from this one: A057501, A074682, A074684, A074686, A074688, A074689, A089866, A120705, A122322, A122331. See also somewhat similar ones: A069774, A071659, A071655, A071657, A072090, A072094, A072092.
Inverse: A074679. Row 17 of A089840. Occurs also in A073200 as row 2156396687 as a(n) = A072796(A073280(A073282(n))). a(n) = A083927(A123497(A057123(n))).
Number of cycles: LEFT(A001683). Number of fixed points: LEFT(A019590). Max. cycle size & LCM of all cycle sizes: A089410 (in range [A014137(n-1)..A014138(n-1)] of this permutation).

Extensions

Description clarified Oct 10 2006

A071660 Permutation of natural numbers induced by the automorphism robl_car_cdr! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 4, 5, 15, 14, 19, 20, 22, 16, 21, 17, 9, 10, 18, 11, 12, 13, 39, 41, 40, 37, 38, 52, 51, 53, 55, 54, 60, 61, 62, 64, 43, 42, 56, 57, 63, 44, 58, 45, 23, 24, 46, 25, 26, 27, 47, 59, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 113, 112, 117, 118, 120
Offset: 0

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Author

Antti Karttunen, May 30 2002

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Inverse permutation: A071659. The car/cdr-flipped conjugate of A071655, i.e. A071660(n) = A057163(A071655(A057163(n))). Cf. also A071657, A071658.

A071656 Permutation of natural numbers induced by the automorphism car_cdr_robl! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 16, 18, 14, 15, 20, 19, 22, 9, 10, 21, 11, 12, 13, 45, 44, 46, 42, 43, 48, 47, 50, 37, 38, 49, 39, 40, 41, 54, 53, 55, 51, 52, 61, 60, 63, 23, 24, 62, 25, 26, 27, 57, 56, 64, 28, 29, 58, 30, 31, 32, 59, 33, 34, 35, 36, 129, 128, 130, 126, 127
Offset: 0

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Author

Antti Karttunen, May 30 2002

Keywords

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Crossrefs

Inverse permutation: A071655. The car/cdr-flipped conjugate of A071659, i.e. A071656(n) = A057163(A071659(A057163(n))). Cf. also A071657, A071658.

A071657 Permutation of natural numbers induced by the automorphism car_robr_cdr! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

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Author

Antti Karttunen, May 30 2002

Keywords

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Crossrefs

Inverse permutation: A071658 and also its car/cdr-flipped conjugate, i.e. A071657(n) = A057163(A071658(A057163(n))). Cf. also A071655, A071656, A071659, A071660.

A071658 Permutation of natural numbers induced by the automorphism cdr_robl_car! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 4, 5, 20, 19, 22, 14, 15, 21, 16, 17, 9, 10, 18, 11, 12, 13, 64, 53, 55, 51, 52, 54, 60, 62, 37, 38, 61, 39, 40, 41, 57, 56, 63, 42, 43, 58, 44, 45, 23, 24, 46, 25, 26, 27, 59, 47, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 162, 191, 192, 154, 155
Offset: 0

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Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071657 and also its car/cdr-flipped conjugate, i.e. A071658(n) = A057163(A071657(A057163(n))). Cf. also A071655, A071656, A071659, A071660.

A072090 Permutation of natural numbers induced by the automorphism cdr_robr_car! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

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Inverse permutation: A072091 and also its car/cdr-flipped conjugate, i.e. A072090(n) = A057163(A072091(A057163(n))). Cf. also A071655-A071660, A072092-A072095.

A072091 Permutation of natural numbers induced by the automorphism car_robl_cdr! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A072090 and also its car/cdr-flipped conjugate, i.e. A072091(n) = A057163(A072090(A057163(n))). Cf. also A071655-A071660, A072092-A072095.
Showing 1-10 of 14 results. Next

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