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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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.

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.

A129611 Signature-permutation of a Catalan automorphism, row 169 of A089840.

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, 49, 50, 54, 55, 61, 63, 64, 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, 133, 134
Offset: 0

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Automorphism *A089859 = ENIPS(*A129611). See the definition given in A122204.

Links

Crossrefs

Inverse: A129612.

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.

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.

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.

A154125 Self-inverse signature permutation of a Catalan bijection: row 83 of A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 06 2009

Keywords

Comments

This bijection of binary trees swaps the left and right subtree of a binary tree, but ONLY if BOTH are nonempty. If either the left or right hand side tree is empty, fixes the tree.
.A...B.C...D.......C...D.A...B.
..\./...\./.........\./...\./..
...x.....x...--->....x.....x...
....\.../.............\.../....
......x.................x......
...............................
((a . b) . (c . d)) -> ((c . d) . (a . b))
or fix, if either the left or right hand side subtree is empty.

Links

Crossrefs

Inverse: A154125. a(n) = A069770(A154126(n)) = A154126(A069770(n)).

A123495 Signature permutation of a nonrecursive Catalan automorphism: row 65518 of table A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Links

Crossrefs

Inverse: A123496. a(n) = A082351(A069770(n)). Row 65518 of A089840. Used to construct automorphism *A082357. Cf. A069770 and A074679.

A123713 Signature permutation of a nonrecursive Catalan automorphism: row 1783367 of table A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Links

Crossrefs

Inverse: A123714. Row 1783367 of A089840. Differs from A089855 for the first time at n=102, where a(n)=103, while A089855(n)=102.

A123714 Signature permutation of a nonrecursive Catalan automorphism: row 1786785 of table A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 11 2006

Keywords

Comments

This automorphism is illustrated below, where letters A, B, C, D, E and F refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.
.............................B...C............F...B......
..............................\./..............\./.......
...............................x...D............x...C....
................................\./..............\./.....
.................................x...E............x...D..
..................................\./.....-->......\./...
..A...B.........C...A..............x...F............x...E
...\./...........\./................\./..............\./.
....x...C...-->...x...B..........()..x............()..x..
.....\./...........\./............\./..............\./...
......x.............x..............x................x....
This is the last multiclause automorphism of total seven opened conses in the table A089840. The next nonrecursive automorphism, A089840[1786786], which consists of a single seven-node clause, swaps the first two toplevel elements (of a general plane tree, like *A072796 does), but only if A057515(n) > 6 and in other cases keeps the tree intact.

Links

Crossrefs

Inverse: A123713. Row 1786785 of A089840. Differs from A089857 for the first time at n=102, where a(n)=106, while A089857(n)=102.
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