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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A085163 Permutation of natural numbers induced by the Catalan bijection gma085163 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 18, 21, 15, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 56, 51, 53, 47, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 46, 49, 38, 32, 59, 58, 62, 40, 29, 55, 35, 43, 27, 50, 63, 52, 41, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085164. a(n) = A085161(A057508(n)). Occurs in A073200. Cf. also A085171, A085172. Scheme-function app-to-xrt given in A085203.
Number of fixed points 1, 1, 2, 2, 3, 4, 7, 11, 21, 36, 71, ... in range [A014137(n-1)..A014138(n-1)] of this permutation appears to be A051920 after the initial two ones.

A086430 Permutation of natural numbers induced by the Catalan bijection gma086430 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 18, 12, 10, 15, 20, 21, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 45, 46, 48, 49, 50, 31, 32, 26, 24, 29, 40, 38, 43, 52, 54, 55, 57, 58, 59, 34, 35, 27, 41, 61, 62, 63, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086429. a(n) = A086428(A086428(n)) = A086431(A086429(A086431(n))) = A057164(A085160(A057164(n))) = A086425(A082316(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A089864 Involution of natural numbers induced by the Catalan automorphism gma089864 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 12, 13, 11, 9, 10, 15, 14, 19, 21, 22, 16, 20, 17, 18, 31, 32, 34, 35, 36, 30, 33, 28, 23, 24, 29, 25, 26, 27, 40, 41, 39, 37, 38, 52, 51, 56, 58, 59, 60, 62, 63, 64, 43, 42, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 87, 88, 90, 91, 92, 96, 97, 99
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This "Catalan bijection" effects the following transformation on the binary trees (labels A,B,C,D refer to arbitrary subtrees located on those nodes and () stands for a terminal node.)
.A..B.C..D.....B..A.D..C.......B...C.......C...B.......A...B........B...A...
..\./.\./.......\./.\./.........\./.........\./.........\./..........\./....
...x...x....-->..x...x.......()..x..-->..()..x...........x..()...-->..x..().
....\./...........\./.........\./.........\./.............\./..........\./..
.....x.............x...........x...........x...............x............x...
i.e. we apply A069770 (that is, the corresponding automorphism) both to the left and right subtree of a binary tree and fix both the empty tree and the tree of one internal node.

Examples

			To obtain this signature permutation, we apply these transformations to the binary trees as encoded and ordered by A014486 and for each n, a(n) will be the position of the tree to which the n-th tree transforms to, as follows:
...................one tree of one internal........2 trees of 2 internal nodes
..empty tree.........(non-leaf) node.................................
........................................................\/.......\/..
......x......................\/........................\/.........\/.
n=....0......................1..........................2..........3.
a(n)=.0......................1..........................2..........3.(all these trees are fixed by this transformation)
however, the next 5 trees, with 3 internal nodes, in range [A014137[2], A014138[2]] = [4,8] change as follows:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
....................|.........................
....................|.........................
....................V.........................
......\/.........\/.............\/.........\/.
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
a(n)=..5........4........6........8........7..
thus we obtain the first nine terms of this sequence: 0,1,2,3,5,4,6,8,7,...

Links

Crossrefs

a(n) = A089859(A089859(n)) = A089863(A089863(n)). Row 1654694 of A089840.
Number of cycles: A089402. Number of fixed points: A089408. Max. cycle size & LCM of all cycle sizes: A046698 (in range [A014137(n-1)..A014138(n-1)] of this permutation).

A089865 Permutation of natural numbers induced by Catalan Automorphism *A089865 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, 20, 21, 16, 22, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 51, 52, 53, 54, 55, 56, 57, 58, 59, 42, 43, 60, 61, 62, 44, 63, 45, 46, 47, 64, 48, 49, 50, 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 *A074679 to the left subtree and keep the right subtree intact.
....B...C.......A...B
.....\./.........\./
..A...x....-->....x...C.................A..().........()..A....
...\./.............\./...................\./....-->....\./.....
....x...D...........x...D.................x...C.........x...C..
.....\./.............\./...................\./...........\./...
......x...............x.....................x.............x....
...............................................................
Compare to A154121.
See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 4207 of A089840. Inverse of A089866. a(n) = A069770(A154121(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

A073431 Number of separate orbits/cycles to which the Catalan bijections A069767/A069768 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 28, 65, 160, 408, 1074, 2898, 7998, 22508, 64426, 187251, 551730, 1645840, 4964876, 15130808, 46545788, 144424944, 451715460
Offset: 0

Views

Author

Antti Karttunen, Jul 31 2002

Keywords

Crossrefs

Occurs for first time in A073201 as row 6 (and 8). Column sums of the square array A074079/Row sums of the triangle A074080.

Programs

  • Maple
    A073431 := proc(n) local i; (1/2^n) * add((2^(n-i))*A073346bi(n,i),i=0..n); end;

Formula

a(0)=1, a(n) = (1/(2^(n-1))) * Sum_{i=1..(2^(n-1))} (Sum_{j=0..A007814(i)} A073346(n, j)) = (1/(2^(n-2))) * Sum_{i=1..(2^(n-1))} A073346(n, A007814(i)) - 1 = (1/2^n) * Sum_{i=0..n} (2^(n-i))*A073346(n, i) = Sum_{i=0..n} A074079(n, i)

A082335 Permutation of natural numbers induced by the Catalan bijection gma082335 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees left, if possible, otherwise reflects them with the Catalan bijection A057163.

Links

Crossrefs

Inverse of A082336. Cf. also A074679-A074680, A082349-A082350.
Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082336 Permutation of natural numbers induced by the Catalan bijection gma082336 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 22, 21, 20, 18, 17, 9, 10, 11, 12, 13, 14, 15, 16, 19, 64, 63, 62, 59, 58, 61, 57, 55, 50, 49, 54, 48, 46, 45, 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, 196, 195, 194, 190, 189
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees right, if possible, otherwise reflects them with the Catalan bijection A057163.

Links

Crossrefs

Inverse of A082335. Cf. also A074679-A074680, A082349-A082350.
Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082340 Permutation of natural numbers induced by the Catalan bijection gma082340 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082339. Occurs in A073200 as row 1798. Cf. also A072797, A082337-A082341.
Max. cycle size and LCMs of cycle sizes: A016116 (to be checked). (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082346 Permutation of natural numbers induced by the Catalan bijection gma082346 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 4, 5, 17, 18, 20, 22, 21, 16, 19, 14, 9, 10, 15, 11, 12, 13, 45, 46, 48, 50, 49, 54, 55, 61, 64, 63, 57, 62, 58, 59, 44, 47, 53, 60, 56, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 129, 130, 132, 134, 133
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082345. Occurs in A073200 as row 88. Cf. also A069768, A073288-A073289, A082347-A082348.

A082349 Permutation of natural numbers induced by the Catalan bijection gma082349 acting on the parenthesizations encoded by A014486/A063171.

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, 10, 9, 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, 29, 26, 27, 64, 28, 25, 24, 23, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees left, if possible, otherwise applies Catalan bijection A069767.

Links

Crossrefs

Inverse of A082350. Cf. also A074679-A074680, A082335-A082336.
Number of cycles: A073193 (to be checked). Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).
Previous Showing 41-50 of 142 results. Next

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