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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A069771 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes180 acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 7, 5, 6, 4, 8, 9, 14, 21, 18, 13, 10, 15, 20, 17, 12, 19, 16, 11, 22, 45, 54, 31, 26, 40, 44, 53, 30, 25, 39, 63, 59, 50, 36, 46, 55, 32, 27, 41, 42, 51, 28, 23, 37, 62, 58, 49, 35, 43, 52, 29, 24, 38, 61, 57, 48, 34, 60, 56, 47, 33, 64, 65, 79, 107, 121, 149, 170
Offset: 0

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Author

Antti Karttunen, Apr 16 2002

Keywords

Comments

This automorphism rotates by 180 degrees the interpretation n (the non-crossing handshakes) of Stanley's exercise 19.

Links

Crossrefs

Cf. A057501, A069772, A069888, A069889.

A126315 Signature-permutation of a Catalan automorphism: composition of A069771 and A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 5, 4, 7, 22, 19, 15, 16, 10, 13, 21, 12, 11, 20, 14, 18, 17, 9, 64, 60, 52, 56, 43, 41, 32, 38, 47, 29, 55, 27, 24, 46, 36, 63, 53, 59, 44, 35, 62, 34, 33, 61, 51, 58, 57, 42, 40, 31, 39, 50, 30, 37, 49, 48, 28, 54, 26, 25, 23, 45, 196, 191, 178, 186, 164
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126316. a(n) = A069771(A125976(n)) = A126290(A069771(n)) = A126313(A057164(n)). The number of cycles, number of fixed points and maximum cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127281, A127282 and A127283. See also the comment at A127280.

A126316 Signature-permutation of a Catalan automorphism: composition of A125976 and A069771.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 8, 4, 22, 13, 17, 16, 14, 19, 11, 12, 21, 20, 10, 18, 15, 9, 63, 35, 62, 61, 34, 59, 32, 55, 52, 29, 45, 44, 42, 37, 56, 30, 53, 51, 28, 50, 27, 41, 64, 36, 31, 58, 57, 54, 47, 25, 39, 60, 33, 26, 49, 48, 40, 24, 46, 43, 38, 23, 196, 120, 106, 148, 78
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126315. a(n) = A125976(A069771(n)) = A069771(A126290(n)) = A057164(A126314(n)).

A129599 Prime-factorization encoded partition code for the Łukasiewicz-word, variant of A129593.

Original entry on oeis.org

1, 3, 25, 25, 343, 35, 35, 343, 35, 14641, 847, 847, 847, 55, 847, 55, 847, 14641, 847, 55, 847, 847, 55, 371293, 24167, 24167, 1573, 1183, 24167, 1183, 1573, 24167, 1183, 1183, 1183, 1183, 65, 24167, 1183, 1183, 1183, 65, 1573, 1183, 24167
Offset: 0

Views

Author

Antti Karttunen, May 01 2007

Keywords

Comments

In addition to all the automorphisms whose signature permutation satisfies the more restricted condition A127301(SP(n)) = A127301(n) for all n, there are also general tree-rotating automorphisms like *A057501, *A057502, *A069771 and *A069772 that satisfy also the condition A129599(SP(n)) = A129599(n) for all n. However, in contrast to A129593 this is not invariant under the automorphism *A072797. A000041(n) distinct values (seem to) occur in each range [A014137(n)..A014138(n)].

Examples

			The terms A079436(5), A079436(6) and A079436(8) are 2010, 2100 and 1110. After adding one to each number except the first one we get 2121, 2211 and 1221, each one which produces partition 1+1+2+2. Converting it to prime-exponents like explained in A129595, we get 2^0 * 3^0 * 5^1 * 7^1 = 35, thus a(5) = a(6) = a(8) = 35.

Links

Crossrefs

Variant: A129593.

Formula

Construction: add one to each number of the Łukasiewicz-word of a general plane tree encoded by A014486(n) (i.e. A079436(n)) except the first number, sort the numbers into ascending order and interpreting it as a partition of a natural number, encode it in the manner explained in A129595.
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