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 1-9 of 9 results.

A130923 Signature permutation of a Catalan automorphism: Inverse FORK-transform of automorphism *A120705.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120705 = (FORK f). See A122201 for the definition of FORK.

Links

Crossrefs

Inverse: A130924. Cf. A130925 & A130926.

A130926 Signature permutation of a Catalan automorphism: Inverse KROF-transform of automorphism *A120705.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120705 = (KROF f). See A122202 for the definition of KROF.

Links

Crossrefs

Inverse: A130925. Cf. A130923 & A130924.

A120707 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 39, 72, 138, 286, 512
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

The number of orbits to which the corresponding automorphisms partition the set of A000108(n) binary trees of n internal nodes.

A120708 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 2, 3, 8, 24, 30, 60, 262, 262, 950, 2508, 4964
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

A120709 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 2, 6, 8, 120, 18480, 314954640, 29650259293200, 806144692105283180937910919057361600, 116647824244848662624579303522985859096483200, 3116996669196650347010384586809853826278378997194387292312487616560888473194736151916402811882584000
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

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

Views

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.

Links

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

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.

A130924 Signature permutation of a Catalan automorphism: Inverse KROF-transform of automorphism *A120706.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 18, 15, 14, 16, 20, 19, 21, 9, 10, 22, 11, 12, 13, 48, 46, 50, 49, 45, 40, 39, 41, 37, 38, 43, 42, 44, 47, 54, 55, 52, 51, 53, 57, 56, 58, 23, 24, 59, 25, 26, 27, 61, 60, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 141, 145, 134, 133, 139
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120706 = (KROF f). See A122202 for the definition of KROF.

Links

Crossrefs

Inverse: A130923. Cf. A130925 & A130926.

A130925 Signature permutation of a Catalan automorphism: Inverse FORK-transform of automorphism *A120706.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 4, 5, 14, 15, 19, 21, 22, 16, 17, 20, 9, 10, 18, 11, 12, 13, 37, 38, 39, 40, 41, 51, 52, 56, 62, 59, 60, 64, 63, 58, 42, 43, 44, 45, 46, 53, 54, 55, 23, 24, 61, 25, 26, 27, 47, 48, 57, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120706 = (FORK f). See A122201 for the definition of FORK.

Links

Crossrefs

Inverse: A130926. Cf. A130923 & A130924.
Showing 1-9 of 9 results.

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