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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A123499 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree left if possible, otherwise apply *A089863.

Original entry on oeis.org

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

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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...............A...B...........A...B.............B...A
....\./.................\./.............\./...............\./.
.A...x........-->........x...C...........x..()...-->...()..x..
..\./.....................\./.............\./...........\./...
...x.......................x...............x.............x....
(a . (b . c)) --> ((a . b) . c) / ((a . b) . ()) --> (() . (b . a))
This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.

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Crossrefs

Inverse: A123500. Row 258 of A089840. Variant of A074679.

A123500 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree right if possible, otherwise apply *A089859.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 21, 22, 20, 17, 18, 9, 10, 11, 12, 13, 14, 15, 16, 19, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 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.
.A...B...............B...C............B...C...........C...B...
..\./.................\./..............\./.............\./....
...x...C....-->....A...x............()..x......-->......x..().
....\./.............\./..............\./.................\./..
.....x...............x................x...................x...
((a . b) . c) --> (a . (b . c)) / (() . (b . c)) --> ((c . b) . ())
This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.

Links

Crossrefs

Inverse: A123499. Row 264 of A089840. Variant of A074680.

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.

A129607 Signature-permutation of a Catalan automorphism: swap the left and right subtree of degree 2 general trees.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Otherwise like automorphism *A072796, except that this involution exchanges the two leftmost subtrees of a general tree ONLY when the degree of the tree is two. Automorphism *A129608 = SPINE(*A129607) = ENIPS(*A129607). See the definitions given in A122203 and A122204.

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Crossrefs

Row 3608 of A089840.

A089841 Main diagonal of A089839. a(n) = A089839(n,n).

Original entry on oeis.org

0, 0, 0, 0, 6, 0, 4, 0, 0, 11, 0, 9, 1653699, 1653578, 1654426, 1654694, 1654312, 1655572, 1654937, 1655787, 1655152, 1654694, 0, 0, 26, 0, 24, 0, 0, 35, 37, 43, 41, 0, 40, 29, 41, 0, 28, 0, 34, 0, 28, 31, 37, 0, 0, 49, 0, 47, 0, 0, 58, 60, 66, 64, 0, 63, 52
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2003

Keywords

Comments

When a(n) is 0, the corresponding row in A089840 (referred here as A089840[n]) is an involution.

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Previous Showing 81-85 of 85 results.

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