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 1-5 of 5 results.

A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 11, 14, 9, 17, 18, 10, 20, 21, 22, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 45, 46, 24, 48, 49, 50, 29, 38, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 87, 88, 90, 91, 92, 96, 97, 99
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C...........C...A
....\./.............\./
.A...x....-->....B...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (b . (c . a)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this rotates car, cadr and cddr of an S-exp
if its length > 1, otherwise keeps it intact.
Note that the first clause corresponds to generator C of Thompson's groups T and V.
(Cf. also A072796, A074679 and A154121 for other related generators).
See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

Links

Crossrefs

Inverse of A089853. a(n) = A089850(A072796(n)) = A057163(A089857(A057163(n))). Row 4 of A089840.
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

The new mail-address, further comments and constructive implementation of Scheme-function (*A089851) added by Antti Karttunen, Jun 04 2011

A089853 Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
...B...C...........A...B
....\./.............\./
.A...x....-->....C...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (c . (a . b)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this automorphism rotates car, cadr and cddr of an S-exp if its length > 1.
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Inverse of A089851. a(n) = A072796(A089850(n)) = A057163(A089855(A057163(n))). Row 6 of A089840.
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089853) added by Antti Karttunen, Jun 04 2011

A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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, 72
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...........B...C
..\./.............\./
...x...C....-->....x...A...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . c) . a) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp if possible, i.e., if car-side is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Inverse of A089857. a(n) = A089860(A069770(n)) = A069770(A074680(n)) = A057163(A089853(A057163(n))). Row 9 of A089840.
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089855) added by Antti Karttunen, Jun 04 2011

A089857 Permutation of natural numbers induced by Catalan Automorphism *A089857 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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, 72
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

.A...B...........C...A
..\./.............\./
...x...C....-->....x...B...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((c . a) . b) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp, i.e., if car-side is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 11 of A089840. Inverse of A089855. a(n) = A074679(A069770(n)) = A069770(A089862(n)) = A057163(A089851(A057163(n))).
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089857) added by Antti Karttunen, Jun 04 2011

A089847 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089851/A089853.

Original entry on oeis.org

1, 1, 2, 3, 8, 24, 72, 231, 764, 2574, 8840, 30796, 108528, 386308, 1386756, 5014575, 18249120, 66786738, 245642760, 907513530, 3366215688, 12531502620, 46805032560, 175341580596, 658674458208, 2480584898700, 9363748625972
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

Programs

Formula

a(n) = (1/3 * A000108(n)) + (2/3 * A089848(n))
Showing 1-5 of 5 results.

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