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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A089860 Permutation of natural numbers induced by Catalan automorphism *A089860 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 4, 5, 21, 22, 19, 14, 15, 20, 16, 17, 9, 10, 18, 11, 12, 13, 58, 59, 62, 63, 64, 56, 60, 51, 37, 38, 52, 39, 40, 41, 57, 61, 53, 42, 43, 54, 44, 45, 23, 24, 46, 25, 26, 27, 55, 47, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 170, 171, 174, 175, 176, 184
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...-->... .x...B...............A..().........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
(a . (b . c)) --> ((c . a) . b) ___ (a . ()) --> (() . a)
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 16 of A089840. Inverse of A089862. a(n) = A089855(A069770(n)) = A069770(A089851(n)) = A069770(A074680(A069770(n))) = A057163(A089862(A057163(n))).
Number of cycles: A001683 (probably, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A089861 Permutation of natural numbers induced by Catalan Automorphism *A089861 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 6, 5, 17, 18, 20, 21, 22, 9, 10, 14, 16, 19, 11, 15, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 32, 33, 41, 34, 35, 36, 129, 130, 132, 133, 134, 138
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...............A...C
..\./.................\./
...x...C...-->.....B...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (b . (a . c)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 18 of A089840. Inverse of A089858. a(n) = A089852(A069770(n)) = A069770(A072797(n)) = A057163(A073269(A057163(n))).
Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A089862 Permutation of natural numbers induced by Catalan automorphism *A089862 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 6, 4, 17, 18, 20, 21, 22, 12, 13, 15, 16, 19, 11, 14, 9, 10, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 24, 29, 38, 25, 26, 27, 129, 130, 132, 133, 134
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...............C...A
..\./.................\./
...x...C...-->.....B...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (b . (c . a)) __ (() . a) ----> (a . ())
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 20 of A089840. Inverse of A089860. a(n) = A089853(A069770(n)) = A069770(A089857(n)) = A069770(A074679(A069770(n))) = A057163(A089860(A057163(n))). Number of cycles: A001683 (seems to be, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A071660 Permutation of natural numbers induced by the automorphism robl_car_cdr! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 4, 5, 15, 14, 19, 20, 22, 16, 21, 17, 9, 10, 18, 11, 12, 13, 39, 41, 40, 37, 38, 52, 51, 53, 55, 54, 60, 61, 62, 64, 43, 42, 56, 57, 63, 44, 58, 45, 23, 24, 46, 25, 26, 27, 47, 59, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 113, 112, 117, 118, 120
Offset: 0

Views

Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071659. The car/cdr-flipped conjugate of A071655, i.e. A071660(n) = A057163(A071655(A057163(n))). Cf. also A071657, A071658.

A071666 Permutation A057506 applied four times ("^4"), permutation A071662 squared.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 11, 19, 17, 20, 12, 10, 15, 21, 18, 13, 22, 23, 37, 42, 28, 51, 44, 53, 30, 25, 39, 56, 48, 33, 60, 45, 54, 57, 34, 61, 31, 40, 26, 24, 38, 43, 47, 29, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 107, 121, 79, 149, 126
Offset: 0

Views

Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071665 and also its car/cdr-flipped conjugate, i.e. A071666(n) = A057163(A071665(A057163(n))) = A057506(A071664(n)) = A071662(A071662(n)). Cf. also A071662, A071668, A071670.

A071667 Permutation A057505 applied five times ("^5").

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 11, 17, 12, 10, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 39, 54, 34, 38, 37, 63, 56, 57, 40, 42, 59, 43, 50, 36, 33, 47, 48, 29, 28, 58, 44, 49, 35, 30, 46, 32, 27, 25, 45, 31, 26, 24, 23, 196, 191, 192, 178, 177
Offset: 0

Views

Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071668 and also its car/cdr-flipped conjugate, i.e. A071667(n) = A057163(A071668(A057163(n))) = A057505(A071665(n)). Cf. also A071661, A071663, A071669.

A074686 Permutation of natural numbers induced by the Catalan bijection gmA074686! acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Links

Crossrefs

Inverse of A074685. a(n) = A057163(A074690(A057163(n))). Occurs in A073200.

A089856 Involution of natural numbers induced by Catalan Automorphism *A089856 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 10, 11, 12, 13, 21, 22, 20, 17, 18, 19, 16, 14, 15, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 38, 52, 43, 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...........C...B
..\./.............\./
...x...C....-->....x...A...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((c . b) . a) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism swaps caar 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

Row 10 of A089840. a(n) = A073269(A069770(n)) = A069770(A073270(n)) = A057163(A089852(A057163(n))).
Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089856) added by Antti Karttunen, Jun 04 2011

A218788 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the right side" construction.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 3, 0, 4, 0, 0, 1, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 4, 0, 0, 2, 0, 37, 0, 0, 110, 0, 3, 0, 1, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 4, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2012

Keywords

Comments

"Tendrils" of the beanstalk are the finite side-trees sprouting from its infinite trunk (see A179016) at the numbers given by A213730.

Examples

			A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25).
When we construct a binary tree from this in such a fashion that the larger numbers go to the left, we obtain:
..........
29...28...
..\./.....
...25..24.
....\./...
.....22...
..........
and the binary tree
.......
.\./...
..*....
...\./.
....*..
.......
is located as A014486(3) in the normal encoding order of binary trees, thus a(9)=3.

Links

Crossrefs

These are the mirror-images of binary trees given in A218787, i.e. a(n) = A057163(A218787(n)). A218786 gives the sizes of these trees. Cf. A072764, A218610, A218611.

A071662 Permutation A057506 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 12, 16, 21, 10, 18, 11, 14, 20, 13, 15, 19, 22, 23, 45, 31, 44, 58, 26, 49, 30, 42, 57, 35, 43, 56, 63, 24, 46, 32, 47, 59, 25, 48, 28, 37, 54, 34, 40, 53, 62, 27, 50, 29, 38, 55, 33, 39, 51, 61, 36, 41, 52, 60, 64, 65, 129, 87, 128, 170, 73
Offset: 0

Views

Author

Antti Karttunen, May 30 2002

Keywords

Links

Crossrefs

Inverse permutation: A071661 and also its car/cdr-flipped conjugate, i.e. A071662(n) = A057163(A071661(A057163(n))) = A057506(A057506(n)). Cf. also A071664, A071666, A071668, A071670.
Previous Showing 71-80 of 167 results. Next

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