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.

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A123498 Signature permutation of a nonrecursive Catalan automorphism: row 1654249 of table A089840.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 10, 17, 16, 18, 11, 20, 12, 14, 15, 13, 19, 21, 22, 23, 24, 25, 26, 27, 45, 46, 44, 42, 43, 48, 47, 49, 50, 28, 29, 54, 53, 55, 30, 57, 31, 37, 38, 32, 39, 40, 41, 33, 61, 34, 51, 52, 35, 56, 58, 59, 36, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 0

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Links

Crossrefs

Inverse: A123497. Row 1654249 of A089840. Used to construct automorphism *A123502. A074679(n) = A083927(a(A057123(n))).

A123499 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree left if possible, otherwise apply *A089863.

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, 9, 10, 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, 28, 23, 24, 64, 29, 25, 26, 27, 107, 108, 109, 110, 111
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...............A...B...........A...B.............B...A
....\./.................\./.............\./...............\./.
.A...x........-->........x...C...........x..()...-->...()..x..
..\./.....................\./.............\./...........\./...
...x.......................x...............x.............x....
(a . (b . c)) --> ((a . b) . c) / ((a . b) . ()) --> (() . (b . a))
This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.

Links

Crossrefs

Inverse: A123500. Row 258 of A089840. Variant of A074679.

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.

A123694 a(n) gives the A089840-index of the nonrecursive Catalan automorphism which is formed from A089840[n] by applying it to the left subtree of a binary tree and leaving the right-hand side subtree intact.

Original entry on oeis.org

0, 7, 91, 92, 93, 94, 95, 114, 115, 116, 117, 118, 4207, 4209, 4211, 4214, 4216, 4299, 4301, 4303, 4305, 4307, 1228, 1229, 1230, 1231, 1232, 1233, 1234, 1235, 1236, 1237, 1238, 1239, 1240, 1241, 1242, 1243, 1244, 1245, 1246, 1247, 1248, 1249, 1250, 1347
Offset: 0

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution). See also the comments at A122200.

Examples

			When A089840[1] = A069770 (swap binary tree sides) is applied to the left subtree of a binary tree, we get A089840[7] = A089854, thus a(1)=7. When A089840[12] = A074679 is applied to the left subtree of a binary tree, we get A089840[4207] = A089865, thus a(12)=4207.

Links

A123695 Signature permutation of a nonrecursive Catalan automorphism: row 1653002 of table A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

It is possible to recursively construct more of these kinds of nonrecursive automorphisms, which by default (if A057515(n) > 1) work as *A074679 and otherwise apply the previous automorphism of this construction process (here *A074679 itself) to the left subtree of a binary tree, before the whole tree is swapped with *A069770. Do the associated cycle-count sequences converge to anything interesting?
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........A...B..............................
............................\./..........\./...............................
..B...C.....A...B........A...x............x...C...A..()...............()..A
...\./.......\./..........\./..............\./.....\./.................\./.
A...x....-->..x...C........x..()...-->..()..x.......x..()....-->....()..x..
.\./...........\./..........\./..........\./.........\./.............\./...
..x.............x............x............x...........x...............x....

Links

Crossrefs

Inverse: A123696. Row 1653002 of A089840. Variant of A074679.

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.

A154123 Signature permutation of a Catalan bijection: row 3656 of A089840.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 11, 12, 13, 16, 19, 15, 14, 9, 17, 18, 10, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 44, 47, 36, 53, 56, 60, 39, 40, 41, 42, 51, 43, 37, 23, 45, 46, 24, 48, 49, 50, 52, 38, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 79, 80, 81, 82, 83, 84, 85
Offset: 0

Views

Author

Antti Karttunen, Jan 06 2009

Keywords

Comments

This bijection of binary trees is obtained in the following way. See also comments at A154122.
....C...D.......B...C
.....\./.........\./
..B...x....-->....x...D.................B..().........()..A..
...\./.............\./...................\./....-->....\./...
A...x...........A...x.................A...x.........B...x....
.\./.............\./...................\./...........\./.....
..x...............x.....................x.............x......
.............................................................
That is, we do (a . (b . (c . d))) -> (a . ((b . c) . d))
or (a . (b . ())) --> (b . (() . a)) if the former is not possible.
Note that the first clause corresponds to generator B of Thompson's groups F, T and V. See further comments at A154121.

Links

Crossrefs

Inverse: A154124. Cf. A154121.

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.

A122321 Row 12 of A122283.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2006

Keywords

Comments

The signature-permutation of the automorphism which is derived from the twelfth non-recursive automorphism *A074679 with recursion schema DEEPEN (see A122283 for the definition).

Links

Crossrefs

Inverse: A122322.

A122332 Row 12 of A122284.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2006

Keywords

Comments

The signature-permutation of the automorphism which is derived from the twelfth non-recursive automorphism *A074679 with recursion schema NEPEED (see A122284 for the definition).

Links

Crossrefs

Inverse: A122331.
Previous Showing 21-30 of 36 results. Next

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