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.

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

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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.

A154121 Signature permutation of a Catalan bijection: row 3655 of A089840.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 11, 12, 13, 9, 10, 15, 14, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 23, 24, 36, 25, 26, 27, 39, 40, 41, 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, 79, 80, 81, 82, 83, 84, 85
Offset: 0

Views

Author

Antti Karttunen, Jan 06 2009

Keywords

Comments

This bijection of binary trees can be obtained by applying bijection *A074679 to the right hand side subtree and leaving the left hand side subtree intact:
....C...D.......B...C
.....\./.........\./
..B...x....-->....x...D.................B..().........()..B..
...\./.............\./...................\./....-->....\./...
A...x...........A...x.................A...x.........A...x....
.\./.............\./...................\./...........\./.....
..x...............x.....................x.............x......
.............................................................
Note that the first clause corresponds to generator B of Thompson's groups F, T and V, while *A074679's first clause corresponds to generator A and furthermore, *A089851 corresponds to generator C and *A072796 to generator pi_0 of Thompson's group V. (To be checked: can Thompson's V be embedded in A089840 by using these or some other suitably chosen generators?)
Comment to above: I think now that it is a misplaced hope to embed V in A089840. Instead, it is more probable that Thompson's V is isomorphic to the quotient group A089840/N, where N is a subgroup of A089840 which includes identity (*A001477) and any other bijection (e.g. *A154126) that fixes all large enough trees. For more details, see my "On the connection of A089840 with ..." page. - Antti Karttunen, Aug 23 2012

Links

Crossrefs

Inverse: A154122. a(n) = A069770(A089865(A069770(n))). Cf. A154123, A154126.

A089866 Permutation of natural numbers induced by Catalan Automorphism *A089866 acting on the parenthesizations/binary trees 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, 17, 18, 20, 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, 44, 45, 46, 47, 48, 49, 50, 53, 54, 55, 57, 61, 65, 66, 67, 68, 69
Offset: 0

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

This bijection of binary trees is obtained when we apply bijection *A074680 to the left subtree and keep the right subtree intact.
.A...B...............B...C
..\./.................\./
...x...C....-->....A...x.................()..B.........B...()....
....\./.............\./...................\./....-->....\./.....
.....x...D...........x...D.................x...C.........x...C..
......\./.............\./...................\./...........\./...
.......x...............x.....................x.............x....
................................................................
Compare to A154122.
See "Catalan bijections" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 4299 of A089840. Inverse of A089865. a(n) = A069770(A154122(A069770(n))).
Number of cycles: A089844. Number of fixed-points: A005807 (prepended with two 1's). Max. cycle size: A089410. LCM of cycle sizes: A089845 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

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.
Showing 1-5 of 5 results.

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