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 1-10 of 15 results. Next

A089840 Signature permutations of non-recursive Catalan automorphisms (i.e., bijections of finite plane binary trees, with no unlimited recursion down to indefinite distances from the root), sorted according to the minimum number of opening nodes needed in their defining clauses.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 10, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 14, 13, 12, 8, 7, 6
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2003; last revised Jan 06 2009

Keywords

Comments

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

References

  • A. Karttunen, paper in preparation, draft available by e-mail.

Links

Crossrefs

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A072796, 3: A089850, 4: A089851, 5: A089852, 6: A089853, 7: A089854, 8: A072797, 9: A089855, 10: A089856, 11: A089857, 12: A074679, 13: A089858, 14: A073269, 15: A089859, 16: A089860, 17: A074680, 18: A089861, 19: A073270, 20: A089862, 21: A089863.
Other rows: row 83: A154125, row 169: A129611, row 183: A154126, row 251: A129612, row 253: A123503, row 258: A123499, row 264: A123500, row 3608: A129607, row 3613: A129605, row 3617: A129606, row 3655: A154121, row 3656: A154123,row 3702: A082354, row 3747: A154122, row 3748: A154124, row 3886: A082353, row 4069: A082351, row 4207: A089865, row 4253: A082352, row 4299: A089866, row 65167: A129609, row 65352: A129610, row 65518: A123495, row 65796: A123496, row 79361: A123492, row 1653002: A123695, row 1653063: A123696, row 1654023: A073281, row 1654249: A123498, row 1654694: A089864, row 1654720: A129604,row 1655089: A123497, row 1783367: A123713, row 1786785: A123714.
Tables A122200, A122201, A122202, A122203, A122204, A122283, A122284, A122285, A122286, A122287, A122288, A122289, A122290, A130400-A130403 give various "recursive derivations" of these non-recursive automorphisms. See also A089831, A073200.
Index sequences to this table, giving various subgroups or other important constructions: A153826, A153827, A153829, A153830, A123694, A153834, A153832, A153833.

A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 21, 22, 20, 17, 18, 19, 16, 14, 9, 10, 15, 11, 12, 13, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 23, 24, 38, 25, 26, 27, 52, 43, 39, 28, 29, 40, 30, 31, 32, 41, 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...B
......\./.........\./
...A...x...-->... .x...A...............A..().........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
(a . (b . c)) --> ((c . b) . 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 15 of A089840. Inverse of A089863. a(n) = A089854(A069770(n)) = A069770(A089850(n)). A089864 is the "square" of this permutation.
Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive implementation of Scheme-function (*A089859) added by Antti Karttunen, Jun 04 2011

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

A089863 Permutation of natural numbers induced by Catalan Automorphism *A089863 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 17, 18, 20, 21, 22, 16, 19, 15, 12, 13, 14, 11, 9, 10, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 37, 28, 23, 24, 38, 29, 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...............B...A
..\./.................\./
...x...C...-->.....C...x...............()..A.........A..()..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) --> (c . (b . 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 21 of A089840. Inverse of A089859. a(n) = A089850(A069770(n)) = A069770(A089854(n)). A089864 is the "square" of this permutation.
Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (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

A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 16, 20, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 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...A
..\./.............\./
...x...C....-->....x...C...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . a) . c) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element 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

a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.
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 (*A089854) 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

A130341 Row 3 of A122203.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

The signature-permutation of the Catalan automorphism which is derived from the third non-recursive Catalan automorphism *A089850 with recursion schema SPINE (see A122203 for the definition). This automorphism is also produced when automorphism *A069767 is applied to the right-hand side subtree of the given binary tree, with the left side left intact.

Links

Crossrefs

Inverse: A130342. a(n) = A069767(A069770(n)). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A089404 and A073268. Cf. also A073286.

A154450 Signature permutation of a Catalan bijection: The inverse of A154449.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it descends along the 111... ray, but not starting swapping until at the right-hand side child of the root, leaving the root itself fixed. Specifically, *A154450 = psi(A154440), where the isomorphism psi is given in A153141 (see further comments there).

Links

Crossrefs

Inverse: A154449. a(n) = A069767(A154456(n)) = A057163(A154454(A057163(n))). Cf. A069770, A154452.
Differs from A089850 for the first time at n=31, where a(31)=24, while A089850(31)=23.

A130342 Row 3 of A122204.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 13, 12, 11, 9, 10, 15, 14, 16, 17, 18, 19, 20, 21, 22, 36, 35, 34, 31, 32, 33, 30, 28, 23, 24, 29, 25, 26, 27, 41, 40, 39, 37, 38, 43, 42, 44, 45, 46, 47, 48, 49, 50, 52, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 106, 105, 104, 100, 101
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

The signature-permutation of the Catalan automorphism which is derived from the third non-recursive Catalan automorphism *A089850 with recursion schema ENIPS (see A122204 for the definition). This automorphism is also produced when automorphism *A069768 is applied to the right-hand side subtree of the given binary tree, with the left side left intact.

Links

Crossrefs

Inverse: A130341. a(n) = A069770(A069768(n)). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A089404 and A073268. Cf. also A073287.

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.
Showing 1-10 of 15 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.