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 1-7 of 7 results.

A089423 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336 (and also of A082349/A082350, to be proved).

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 120, 840, 840, 5040, 5040, 55440, 55440, 720720, 720720, 720720, 720720, 24504480, 24504480, 465585120, 465585120
Offset: 0

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Author

Antti Karttunen, Nov 29 2003

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Bisection: A089431.

A089422 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336 (and also of A082349/A082350, to be proved).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38
Offset: 0

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Author

Antti Karttunen, Nov 29 2003

Keywords

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Crossrefs

Cf. A005843.

A089421 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336.

Original entry on oeis.org

1, 1, 1, 2, 3, 9, 22, 71, 217, 729, 2438, 8440, 29414, 104138, 371516, 1337649, 4847637, 17680265, 64823110, 238824212, 883634026
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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A074679 Signature permutation of a Catalan automorphism: Rotate binary tree left if possible, otherwise swap its sides.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 4, 5, 14, 15, 16, 17, 18, 19, 20, 21, 9, 10, 22, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 23, 24, 59, 25, 26, 27, 60, 61, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 107, 108, 109, 110, 111
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.)
...B...C.......A...B
....\./.........\./
.A...x....-->....x...C.................A..().........()..A..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)
That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left 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.
This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

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This automorphism has several variants, where the first clause is same (rotate binary tree to the left, if possible), but something else is done (than just swapping sides), in case the right hand side is empty: A082335, A082349, A123499, A123695. The following automorphisms can be derived recursively from this one: A057502, A074681, A074683, A074685, A074687, A074690, A089865, A120706, A122321, A122332. See also somewhat similar ones: A069773, A071660, A071656, A071658, A072091, A072095, A072093.
Inverse: A074680.
Row 12 of A089840.
Occurs also in A073200 as row 557243 because a(n) = A073283(A073280(A072796(n))). a(n) = A083927(A123498(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)] of this permutation).

Extensions

Description clarified Oct 10 2006

A082336 Permutation of natural numbers induced by the Catalan bijection gma082336 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 22, 21, 20, 18, 17, 9, 10, 11, 12, 13, 14, 15, 16, 19, 64, 63, 62, 59, 58, 61, 57, 55, 50, 49, 54, 48, 46, 45, 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, 196, 195, 194, 190, 189
Offset: 0

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Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees right, if possible, otherwise reflects them with the Catalan bijection A057163.

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Inverse of A082335. Cf. also A074679-A074680, A082349-A082350.
Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082349 Permutation of natural numbers induced by the Catalan bijection gma082349 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

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Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees left, if possible, otherwise applies Catalan bijection A069767.

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Inverse of A082350. Cf. also A074679-A074680, A082335-A082336.
Number of cycles: A073193 (to be checked). Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082350 Permutation of natural numbers induced by the Catalan bijection gma082350 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 22, 21, 20, 17, 18, 9, 10, 11, 12, 13, 14, 15, 16, 19, 64, 63, 62, 58, 59, 61, 57, 54, 45, 46, 55, 48, 49, 50, 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, 196, 195, 194, 189, 190
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees right, if possible, otherwise applies Catalan bijection A069768.

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Crossrefs

Inverse of A082349. Cf. also A074679-A074680, A082335-A082336.
Number of cycles: A073193 (to be checked). Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).
Showing 1-7 of 7 results.

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