A082352 -id:A082352 - cp's OEIS frontend

A082362 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082352.

0, 1, 3, 4, 2, 8, 9, 11, 12, 13, 5, 6, 7, 10, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 38, 39, 40, 41, 14, 15, 16, 17, 18, 19, 20, 21, 24, 28, 29, 30, 33, 37, 64, 65, 67, 68, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 92, 93, 95, 96, 97, 101, 102, 104, 105, 106, 108, 109, 110, 111

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082361. Cf. also A082363-A082364.

a(n) = A082853(A082352(A081291(n))).

A089424 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082351/A082352.

Original entry on oeis.org

1, 1, 2, 3, 6, 18, 48, 151, 478, 1580, 5304, 18220, 63308, 222936, 792436, 2841807, 10265130, 37322728, 136468200, 501523066, 1851418668

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

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

A089425 Least common multiple of all cycle sizes (and also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A082351/A082352.

Original entry on oeis.org

1, 1, 1, 3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

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

Cf. A008585, A008486.

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

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

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.

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

A. Karttunen, C-program for computing the terms of this table. Defines also the order of rows.
A. Karttunen, Prolog-program which illustrates the construction of each row

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.

A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Inverse of A082356. a(n) = A082357(A057163(n)). a(n) = A082363(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.

Differs from A057118 first time at n=42: a(42)=56, while A057118(42)=58.

A082356 Permutation of natural numbers induced by Catalan Automorphism *A082356 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Inverse of A082355. a(n) = A057163(A082358(n)). a(n) = A082364(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.

Differs from A057117 first time at n=56: a(56)=42, while A057117(56)=44.

A082351 Permutation of natural numbers induced by the Catalan bijection gma082351 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082352. a(n) = A057163(A082353(A057163(n))). Cf. also A074679-A074680, A082355-A082356, A082361-A082362.

A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123495. a(n) = A069770(A082352(n)). Row 65796 of A089840. Used to construct automorphism *A082358. Cf. A069770 and A074680.

A082354 Permutation of natural numbers induced by the Catalan bijection gma082354 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082353. a(n) = A057163(A082352(A057163(n))).

cp's OEIS Frontend

A082362 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082352.

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A089424 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A082351/A082352.

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A089425 Least common multiple of all cycle sizes (and also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A082351/A082352.

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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.

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A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

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A082356 Permutation of natural numbers induced by Catalan Automorphism *A082356 acting on the parenthesizations encoded by A014486/A063171.

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A082351 Permutation of natural numbers induced by the Catalan bijection gma082351 acting on the parenthesizations encoded by A014486/A063171.

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A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

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A082354 Permutation of natural numbers induced by the Catalan bijection gma082354 acting on the parenthesizations encoded by A014486/A063171.

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