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.

A089429 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082339/A082340.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 74, 218, 672, 2134, 6964, 23206, 78724, 271014, 944766, 3328872, 11838378, 42441326, 153236900, 556746772, 2034098988
Offset: 0

Views

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

A089430 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A082339/A082340.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 30, 79, 232, 716, 2318, 7714, 26270, 90944, 319146, 1132185, 4053618, 14626828, 53136324, 194174324, 713267792
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of n-node binary trees fixed by the corresponding automorphism(s).

Links

A122203 Signature permutations of SPINE-transformations of non-recursive Catalan automorphisms in table A089840.

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, 6, 5, 3, 2, 1, 0, 8, 5, 5, 4, 5, 3, 2, 1, 0, 9, 4, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 11, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 14, 13, 12
Offset: 0

Views

Author

Antti Karttunen, Sep 01 2006, Jun 06 2007

Keywords

Comments

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "SPINE". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the new right-hand side branch. The associated Scheme-procedures SPINE and !SPINE can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122204.
The recursion scheme SPINE has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.

References

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

Links

Crossrefs

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069767, 2: A057509, 3: A130341, 4: A130343, 5: A130345, 6: A130347, 7: A122282, 8: A082339, 9: A130349, 10: A130351, 11: A130353, 12: A074685, 13: A130355, 14: A130357, 15: A130359, 16: A130361, 17: A057501, 18: A130363, 19: A130365, 20: A130367, 21: A069770. Other rows: row 251: A089863, row 253: A123717, row 3608: A129608, row 3613: A072796, row 65352: A074680, row 79361: A123715.
See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.

Programs

  • Scheme
    (define (SPINE foo) (letrec ((bar (lambda (s) (let ((t (foo s))) (if (pair? t) (cons (car t) (bar (cdr t))) t))))) bar))
(define (!SPINE foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! s) (bar! (cdr s)))) s))) bar!))

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

Views

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.

A082340 Permutation of natural numbers induced by the Catalan bijection gma082340 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, 49, 48, 50, 44, 47, 42, 37, 38, 43, 39, 40, 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 A082339. Occurs in A073200 as row 1798. 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).

A082341 Permutation of natural numbers induced by the Catalan bijection gma082341 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, 21, 19, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 34, 35, 33, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 64, 65, 66, 68, 67, 69
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This is A057163-conjugate of A073285.

Links

Crossrefs

Inverse of A082342. a(n) = A057163(A073285(A057163(n))). Occurs in A073200 as row 1800. Cf. also A072797, A082337-A082339.
Differs from A082326 first time at n=39: a(39)=48, while A082326(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082342 Permutation of natural numbers induced by the Catalan bijection gma082342 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, 21, 19, 20, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 35, 33, 34, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 58, 59, 56, 51, 52, 57, 53, 54, 55, 63, 60, 61, 62, 64, 65, 66, 68, 67, 69
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This is A057163-conjugate of A073284.

Links

Crossrefs

Inverse of A082341. a(n) = A057163(A073284(A057163(n))). Occurs in A073200. Cf. also A072797, A082337-A082339.
Differs from A082325 first time at n=39: a(39)=48, while A082325(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A130987 Signature permutation of a Catalan automorphism: row 8 of A122288.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 20, 21, 19, 22, 23, 24, 26, 25, 27, 28, 29, 30, 31, 32, 33, 35, 34, 36, 45, 46, 49, 48, 50, 44, 47, 42, 37, 38, 43, 40, 39, 41, 54, 55, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 64, 65, 66, 67, 68, 69, 73, 74
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A082339 with recursion scheme KROF.

Links

Crossrefs

Inverse: A130988.

A131169 Signature permutation of a Catalan automorphism: row 8 of A122285.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 20, 19, 21, 22, 23, 24, 26, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 49, 48, 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, 67, 68, 69, 73, 74
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A082339 with recursion scheme ENIPS.

Links

Crossrefs

Inverse: A131170.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.