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.

Previous Showing 21-30 of 32 results. Next

A123500 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree right if possible, otherwise apply *A089859.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 21, 22, 20, 17, 18, 9, 10, 11, 12, 13, 14, 15, 16, 19, 58, 59, 62, 63, 64, 57, 61, 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, 170, 171, 174, 175, 176
Offset: 0

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.
.A...B...............B...C............B...C...........C...B...
..\./.................\./..............\./.............\./....
...x...C....-->....A...x............()..x......-->......x..().
....\./.............\./..............\./.................\./..
.....x...............x................x...................x...
((a . b) . c) --> (a . (b . c)) / (() . (b . c)) --> ((c . b) . ())
This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.

Links

Crossrefs

Inverse: A123499. Row 264 of A089840. Variant of A074680.

A089831 Triangle T(n,m) (read as T(1,1); T(2,1), T(2,2); T(3,1), T(3,2), T(3,3);) Number of distinct non-recursive Catalan Automorphisms whose minimum clause-representation requires examination of n nodes in total, divided into m non-default clauses.

Original entry on oeis.org

1, 10, 0, 115, 10, 0, 1666, 139, 0, 0, 30198, 2570, 0, 0, 0, 665148, 47878, 904, 0, 0, 0, 17296851, 1017174, 20972, 0, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2003

Keywords

Examples

			...... Triangle............................ Row sums
........1........................................1
.......10.......0...............................10
......115......10...0..........................125 = 5^3
.....1666.....139...0....0....................1805 = 5*19^2
....30198....2570...0....0...0...............32768 = 32^3 = 8^5
...665148...47878...904..0...0...0..........713930
.17296851.1017174.20972..0...0...0...0....18334997
T(1,1)=1, as there is just one non-identity, non-recursive Catalan bijection with a single non-default clause opening a single node, namely A089840[1]=A069770.
T(2,1)=10, as there are the following non-recursive Catalan bijections (rows 2-11 of A089840): A072796, A089850, A089851, A089852, A089853, A089854, A072797, A089855, A089856, A089857, whose minimum clause-representation consists of a single non-default clause that opens two nodes.
T(3,2)=10, as there are the following non-recursive Catalan bijections (rows 12-21 of A089840): A074679, A089858, A073269, A089859, A089860, A074680, A089861, A073270, A089862, A089863, whose minimum clause-representation consists of a two non-default clauses with total 3 nodes opened.

Links

Crossrefs

First column: A089833. Row sums: A089832. Row sums excluding the first column: A089834.

A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 5, 6, 17, 18, 20, 21, 22, 9, 10, 11, 12, 13, 16, 19, 14, 15, 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, 44, 47, 53, 56, 60, 42, 51, 37, 38, 43, 52, 39, 40, 41, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Links

Crossrefs

Inverse: A123495. a(n) = A069770(A082352(n)). Row 65796 of A089840. Used to construct automorphism *A082358. Cf. A069770 and A074680.

A123696 Signature permutation of a nonrecursive Catalan automorphism: row 1653063 of table A089840.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 21, 22, 17, 18, 20, 9, 10, 11, 12, 13, 14, 15, 16, 19, 58, 59, 62, 63, 64, 45, 46, 48, 49, 50, 54, 55, 57, 61, 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, 170, 171, 174, 175, 176
Offset: 0

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.
............................B...C.......C...D..............................
.............................\./.........\./...............................
.A...B.............B...C......x...D....B..x............()...C......C..()...
..\./...............\./........\./......\./.............\./.........\./....
...x...C..-->....A...x......()..x...-->..x..().......()..x....-->....x..().
....\./...........\./........\./..........\./.........\./.............\./..
.....x.............x..........x............x...........x...............x...
See the comments at A123695.

Links

Crossrefs

Inverse: A123695. Row 1653063 of A089840. Variant of A074680.

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.

A154124 Signature permutation of a Catalan bijection: row 3748 of A089840.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 7, 8, 16, 19, 9, 10, 11, 15, 14, 12, 17, 18, 13, 20, 21, 22, 44, 47, 53, 56, 60, 23, 24, 25, 26, 27, 28, 29, 30, 33, 43, 52, 37, 38, 39, 40, 42, 31, 45, 46, 32, 48, 49, 50, 41, 51, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 128, 131, 137, 140, 144
Offset: 0

Views

Author

Antti Karttunen, Jan 06 2009

Keywords

Comments

This bijection of binary trees is obtained in the following way. (Inverse of Thompson's B!):
.B...C...............C...D
..\./.................\./
...x...D....-->....B...x.................()..C ........A...()
....\./.............\./...................\./....-->....\./...
.A...x...........A...x.................A...x.........C...x....
..\./.............\./...................\./...........\./.....
...x...............x.....................x.............x......
..............................................................
That is, (a . ((b . c) . d)) -> (a . (b . (c . d)))
or (a . (() . c)) -> (c . (a . ())) if the former is not possible.

Links

Crossrefs

Inverse: A154123. Cf. A154122.

A085168 Inverse permutation to A085167.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 8, 6, 5, 17, 9, 18, 14, 10, 20, 11, 21, 16, 12, 22, 19, 15, 13, 45, 23, 46, 37, 24, 48, 25, 49, 42, 26, 50, 51, 38, 27, 54, 28, 55, 39, 29, 57, 30, 58, 44, 31, 59, 47, 40, 32, 61, 33, 62, 53, 34, 63, 56, 43, 35, 64, 60, 52, 41, 36, 129, 65, 130, 107, 66
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085167. a(n) = A069770(A085160(n)). Occurs in A073200. Cf. also A074679, A074680, A085203.

A122322 Row 17 of A122284.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2006

Keywords

Comments

The signature-permutation of the automorphism which is derived from the seventeenth non-recursive automorphism *A074680 with recursion schema NEPEED (see A122284 for the definition).

Links

Crossrefs

Inverse: A122321.

A122331 Row 17 of A122283.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2006

Keywords

Comments

The signature-permutation of the automorphism which is derived from the seventeenth non-recursive automorphism *A074680 with recursion schema DEEPEN (see A122283 for the definition).

Links

Crossrefs

Inverse: A122332.

A129609 Signature-permutation of a Catalan automorphism, row 65167 of A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Automorphism *A074679 = ENIPS(*A129609). See the definition given in A122204.

Links

Crossrefs

Inverse: A129610.
Previous Showing 21-30 of 32 results. Next

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

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