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.

A089406 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082345/A082346.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 37, 95, 238, 624, 1650, 4478, 12356, 34756, 99313, 288145, 847366, 2523092, 7597834, 23117382, 71007196
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.

Links

A069767 Signature-permutation of Catalan bijection "Knick".

Original entry on oeis.org

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

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Author

Antti Karttunen, Apr 16 2002; entry revised Dec 20 2008

Keywords

Comments

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153141. See further comments there.
This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.
Note: the name in Finnish is "Niks".

References

  • A. Karttunen, paper in preparation.

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Crossrefs

Inverse permutation: "Knack", A069768. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073290, A073292, A073294, A073296, A073298.
In range [A014137(n-1)..A014138(n-1)] of this permutation, the number of cycles is A073431, number of fixed points: A036987 (Fixed points themselves: A084108), Max. cycle size & LCM of all cycle sizes: A011782. See also: A074080.
A127302(a(n)) = A127302(n) for all n. a(n) = A057508(A057161(n)) = A057161(A069769(n)).
Row 1 of A122203 and A122287, row 15 of A122286 and A130403, row 6 of A073200.
See also bijections A073286, A082345, A082348, A082349, A130341.

A130403 Signature permutations of SPINE-transformations of A057163-conjugates of Catalan automorphisms in table A122204.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 4, 7, 5, 4, 3, 2, 1, 0, 9, 5, 6, 6, 5, 4, 3, 2, 1, 0, 10, 17, 8, 8, 8, 5, 4, 3, 2, 1, 0, 11, 18, 9, 7, 6, 8, 5, 5, 3, 2, 1, 0, 12, 20, 10, 9, 7, 7, 7, 4, 4, 3, 2, 1, 0, 13, 21, 12, 10, 9, 6
Offset: 0

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Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122204 with the recursion scheme "SPINE", i.e. row n is obtained as SPINE(A057163 o ENIPS(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130402. This table contains also all the rows of A122203 and A089840.

Crossrefs

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082345, 2: A130936, 3: A073288, 4: A130942, 5: A130940, 6: A130938, 7: A130944, 8: A130946, 9: A130952, 10: A130950, 11: A130948, 12: A057161, 13: A130962, 14: A130964, 15: A069767, 16: A130966, 17: A074688, 18: A130954, 19: A130956, 20: A130960, 21: A130958, Other rows: 169: A069770, 3617: A082339, 65167: A057501.
Cf. also tables A089840, A122201-A122204, A122285-A122286, A130400-A130401.
Cf. As a sequence differs from A130403 for the first time at n=92, where a(n)=21, while A130403(n)=22.

A082346 Permutation of natural numbers induced by the Catalan bijection gma082346 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, 22, 21, 16, 19, 14, 9, 10, 15, 11, 12, 13, 45, 46, 48, 50, 49, 54, 55, 61, 64, 63, 57, 62, 58, 59, 44, 47, 53, 60, 56, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 129, 130, 132, 134, 133
Offset: 0

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Author

Antti Karttunen, Apr 17 2003

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Crossrefs

Inverse of A082345. Occurs in A073200 as row 88. Cf. also A069768, A073288-A073289, A082347-A082348.

A154455 Signature permutation of a Catalan bijection induced by generator "b" of the leftward recursing instance of Basilica group wreath recursion: a = (b,1), b = s(a,1).

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, 26, 27, 43, 52, 39, 28, 29, 41, 33, 35, 36, 40, 30, 34, 31, 32, 129, 130, 132, 133, 134
Offset: 0

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Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it returns back toward the root, after descending down to the leftmost tip of the tree along the 000... ray, so that the last vertex whose descendants are swapped is the root node of the tree. Specifically, *A154455 = psi(A154445), where the isomorphism psi is given in A153141 (see further comments there).

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Crossrefs

Inverse: A154456. a(n) = A154449(A069767(n)) = A057163(A154451(A057163(n))). Cf. A069770, A154453.
Differs from A082345 for the first time at n=49, where a(49)=26, while A082345(49)=27. Differs from A122327 for the first time at n=26, where a(26)=49, while A122327(26)=50. Differs from A129612 for the first time at n=195, where a(195)=92, while A129612(195)=91.

A082347 Permutation of natural numbers induced by the Catalan bijection gma082347 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

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Author

Antti Karttunen, Apr 17 2003

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Inverse of A082348. Occurs in A073200 as row 86. Cf. also A069768, A073288-A073289, A082345-A082346.

A082348 Permutation of natural numbers induced by the Catalan bijection gma082348 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

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Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082347. Occurs in A073200 as row 68. Cf. also A069767, A073288-A073289, A082345-A082346.

A129612 Signature-permutation of a Catalan automorphism, row 251 of A089840.

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, 26, 27, 43, 52, 39, 28, 29, 41, 33, 35, 36, 40, 30, 34, 31, 32, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Automorphism *A089863 = SPINE(*A129612). See the definition given in A122203.

Links

Crossrefs

Inverse: A129611. Differs from A082345 for the first time at n=49, where A082345(49)=27, while a(49)=26.
Showing 1-8 of 8 results.

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