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 1-6 of 6 results.

A127384 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutations A127379/A127380 and A127381/A127382.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 62, 179, 543, 1705, 5482, 18056, 60634, 206824
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

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A127383.

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

Examples

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Links

Crossrefs

Cf A127301, A153835, A153829.

Formula

a(n) = A127301(A057123(n)).
Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A127379 Signature-permutation of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

It's much easier to implement Callan's 2006 bijection for S-expressions if one considers a mirror-image of the graphical description given by Callan (on page 3). Then this automorphism is just RIBS-transformation (explained in A122200) of the automorphism A127377 and Callan's original variant A127381 is obtained as A057164(a(A057164(n))).

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Inverse: A127380. a(n) = A057164(A127381(A057164(n))). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127384 and A086625 shifted once right. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, ... A127302(a(n)) = A127302(n) holds for all n. A127388 shows a variant which is an involution.
Differs from A073289 and A122349 for the first time at n=54, where a(n)=54, while A073289(54) = A122349(54) = 61.

A127378 Signature-permutation of a Catalan automorphism, inverse of A127377.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Used to construct A127380.

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Inverse: A127377. A127302(a(n)) = A127302(n) holds for all n.

A127388 Signature-permutation of a Catalan automorphism, a self-inverse variant of A127379.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This automorphism is RIBS-transformation (explained in A122200) of the automorphism A127387.

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Crossrefs

The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this involution are given by A127386 and A086625 shifted once right (this automorphism has the same fixed points as A127379/A127380). A127302(a(n)) = A127302(n) holds for all n.

A127382 Signature-permutation of the inverse of Callan's 2006 bijection (A127381) on Dyck Paths.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 16 2007

Keywords

Links

Crossrefs

Inverse: A127381. a(n) = A057164(A127380(A057164(n))).
Showing 1-6 of 6 results.

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