A089851 -id:A089851 - cp's OEIS frontend

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.

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

Antti Karttunen, Dec 05 2003

			...... 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.

A. Karttunen, C-program for computing the initial terms of this sequence
A. Karttunen, Prolog-program which illustrates the construction of non-recursive Catalan bijections with clause-representations
A. Karttunen, Catalan Automorphisms

First column: A089833. Row sums: A089832. Row sums excluding the first column: A089834.

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

Antti Karttunen, Jan 06 2009

nonn

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.

A. Karttunen, Table of n, a(n) for n = 0..2055
J. W. Cannon, W. J. Floyd, and W. R. Parry, Notes on Richard Thompson's Groups F and T
J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathématique, Vol. 42 (1996), pp. 215-256.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A154124. Cf. A154121.

A130381 Signature permutation of a Catalan automorphism: row 4 of A130400.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 9, 14, 10, 18, 20, 11, 17, 21, 22, 32, 34, 31, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 24, 25, 37, 42, 51, 23, 38, 27, 49, 50, 29, 55, 57, 61, 28, 39, 26, 45, 54, 30, 46, 59, 62, 33, 48, 58, 63, 64, 91, 92, 97, 99, 103, 87

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

The signature-permutation of the Catalan automorphism which is derived from the fourth non-recursive Catalan automorphism *A089851 with recursion schema INORDER (see A130400 for the definition).

Inverse: A130382.

A130386 Signature permutation of a Catalan automorphism: row 4 of A130401.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 13, 15, 12, 16, 19, 11, 14, 9, 18, 20, 10, 17, 21, 22, 36, 41, 35, 43, 52, 34, 40, 31, 53, 44, 32, 47, 56, 60, 33, 39, 30, 42, 51, 28, 37, 23, 50, 55, 24, 49, 57, 61, 29, 38, 25, 48, 54, 26, 45, 59, 62, 27, 46, 58, 63, 64, 106, 120, 105, 125, 153

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

The signature-permutation of the Catalan automorphism which is derived from the fourth non-recursive Catalan automorphism *A089851 with recursion schema REDRONI (see A130401 for the definition).

Inverse: A130385.

A089842 Order of each element (row) in A089840, 0 if not finite.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4

Offset: 0

Antti Karttunen, Dec 05 2003

nonn

If a(n) is nonzero, then the n-th non-recursive Catalan Automorphism in A089840 does not have orbits (cycles) larger than that and the corresponding LCM-sequence will set to a constant sequence a(n),a(n),a(n),a(n),... E.g. A089840[4] = A089851 is obtained by rotating three subtrees cyclically and its LCM-sequence begins as 1,1,1,3,3,3,3,3,3,3,3,... (a(4)=3). Similarly, A089840[15] = A089859, whose LCM-sequence begins as 1,1,2,4,4,4,4,4,4,4,4,.... (a(15)=4). In contrast, the Max. cycle and LCM-sequence (A089410) of A089840[12] (= A074679) exhibits genuine growth, thus a(12)=0.

A. Karttunen, C-program for computing the initial terms of this sequence

Note that the terms 1-23 of A060131: 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4 repeat here at positions [22..44], [45..67], [68..90], [91..113], [114..136].

A122305 Row 4 of A122283.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 11, 14, 9, 18, 20, 10, 17, 21, 22, 32, 34, 31, 35, 36, 40, 41, 43, 47, 53, 52, 44, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 49, 50, 24, 55, 57, 61, 29, 38, 25, 48, 54, 26, 45, 59, 62, 27, 46, 58, 63, 64, 91, 92, 97, 99, 103, 90

Offset: 0

Antti Karttunen, Sep 01 2006

nonn

The signature-permutation of the automorphism which is derived from the fourth non-recursive automorphism *A089851 with recursion schema DEEPEN (see A122283 for the definition).

Index entries for signature-permutations of Catalan automorphisms

Inverse: A122306.

A122310 Row 4 of A122284.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 01 2006

nonn

The signature-permutation of the automorphism which is derived from the fourth non-recursive automorphism *A089851 with recursion schema NEPEED (see A122284 for the definition).

Index entries for signature-permutations of Catalan automorphisms

Inverse: A122309.

A122343 Row 4 of A122201.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 9, 14, 10, 18, 20, 11, 17, 21, 22, 32, 34, 31, 35, 36, 40, 41, 43, 47, 53, 52, 44, 56, 60, 24, 25, 37, 42, 51, 23, 38, 26, 49, 50, 27, 55, 57, 61, 28, 39, 29, 45, 54, 30, 46, 59, 62, 33, 48, 58, 63, 64, 91, 92, 97, 99, 103, 87

Offset: 0

Antti Karttunen, Sep 01 2006

nonn

The signature-permutation of the automorphism which is derived from the fourth non-recursive automorphism *A089851 with recursion schema FORK (see A122201 for the definition).

Index entries for signature-permutations of Catalan automorphisms

Inverse: A122344.

A122348 Row 4 of A122202.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 13, 15, 12, 16, 19, 11, 14, 9, 18, 20, 10, 17, 21, 22, 36, 41, 35, 43, 52, 34, 40, 31, 47, 53, 32, 44, 56, 60, 33, 39, 30, 42, 51, 28, 37, 24, 50, 55, 25, 49, 57, 61, 29, 38, 23, 48, 54, 26, 45, 59, 62, 27, 46, 58, 63, 64, 106, 120, 105, 125, 153

Offset: 0

Antti Karttunen, Sep 01 2006

nonn

The signature-permutation of the automorphism which is derived from the fourth non-recursive automorphism *A089851 with recursion schema KROF (see A122202 for the definition).

Index entries for signature-permutations of Catalan automorphisms

Inverse: A122347.

cp's OEIS Frontend

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.

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A154123 Signature permutation of a Catalan bijection: row 3656 of A089840.

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A130381 Signature permutation of a Catalan automorphism: row 4 of A130400.

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A130386 Signature permutation of a Catalan automorphism: row 4 of A130401.

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A089842 Order of each element (row) in A089840, 0 if not finite.

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A122305 Row 4 of A122283.

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A122310 Row 4 of A122284.

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A122343 Row 4 of A122201.

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A122348 Row 4 of A122202.

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