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 1-5 of 5 results.

A154452 Signature permutation of a Catalan bijection: The inverse of A154451.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it descends along the 111... ray, starting swapping already at the root. Specifically, *A154452 = psi(A154442), where the isomorphism psi is given in A153141 (see further comments there).

Links

Crossrefs

Inverse: A154451. a(n) = A069767(A154454(n)) = A057163(A154456(A057163(n))). Cf. A069770, A154450.

A154449 Signature permutation of a Catalan bijection induced by generator "a" of the rightward recursing instance of Basilica group wreath recursion: a = (1,b), b = s(1,a).

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it returns back toward the root, after descending down to the rightmost tip of the tree along the 111... ray, so that the last vertex whose descendants are swapped, is the right-hand side child of the root and the root itself is fixed. Specifically, *A154449 = psi(A154439), where the isomorphism psi is given in A153141 (see further comments there).

Links

Crossrefs

Inverse: A154450. a(n) = A154455(A069768(n)) = A057163(A154453(A057163(n))). Cf. A154451.

A154441 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (1,b), b = s(1,a), starting from the active (swapping) state b.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 12, 13, 14, 15, 8, 9, 11, 10, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 22, 23, 20, 21, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 40, 41, 43, 42, 96, 97, 98, 99, 100, 101, 102
Offset: 0

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Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This permutation is induced by the Basilica group generating wreath recursion a = (1,b), b = s(1,a) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on the page 40 of Bartholdi and Virag paper, starting from the active (switching) state b and rewriting bits from the second most significant bit to the least significant end.

Examples

			Starting from the second most significant bit, we continue complementing every second bit (in this case, starting from the second most significant bit), as long as the first zero is encountered, which is also complemented if its distance to the most significant bit is odd, after which the remaining bits are left intact. E.g. 121 = 1111001 in binary. Complementing its second and fourth most significant bits (positions 5 & 3) and stopping at the first zero-bit at position 2 (which is not complemented, as its distance to the msb is 6), we obtain "10100.." after which the rest of the bits stay same, so we get 1010001, which is 81's binary representation, thus a(121)=81. On the other hand, 125 = 1111101 in binary and the transducer complements the bits at positions 5, 3 and also the first zero at the position 1 (because at odd distance from the msb), yielding 101011., after which the remaining bit stays same, thus we get 1010111, which is 87's binary representation, thus a(125)=87.

References

  • R. I. Grigorchuk and A. Zuk, Spectral properties of a torsion free weakly branch group defined by a three state automaton, Contemporary Mathematics 298 (2002), 57--82.

Links

Crossrefs

Inverse: A154442. a(n) = A154443(A153142(n)) = A054429(A154445(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154451 in the group of Catalan bijections.

Extensions

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A154453 Signature permutation of a Catalan bijection induced by generator "a" of the leftward recursing instance of Basilica group wreath recursion: a = (b,1), b = s(a,1).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it returns back toward the root, after descending down to the leftmost tip of the tree along the 000... ray, so that the last vertex whose descendants are swapped, is the left-hand side child of the root and the root itself is fixed. Specifically, *A154453 = psi(A154443), where the isomorphism psi is given in A153141 (see further comments there).

Links

Crossrefs

Inverse: A154454. a(n) = A154451(A069767(n)) = A057163(A154449(A057163(n))). Cf. A154455.
Differs from its inverse A154454 for the first time at n=49, where a(49)=63, while A154454(49)=64. Differs from A089854 for the first time at n=63, where a(63)=50, while A089854(63)=49. Differs from A131173 for the first time at n=26, where a(26)=26, while A131173(26)=27.

A154455 Signature permutation of a Catalan bijection induced by generator "b" of the leftward recursing instance of Basilica group wreath recursion: a = (b,1), b = s(a,1).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 17 2009

Keywords

Comments

This automorphism of rooted plane binary trees switches the two descendant trees for every other vertex as it returns back toward the root, after descending down to the leftmost tip of the tree along the 000... ray, so that the last vertex whose descendants are swapped is the root node of the tree. Specifically, *A154455 = psi(A154445), where the isomorphism psi is given in A153141 (see further comments there).

Links

Crossrefs

Inverse: A154456. a(n) = A154449(A069767(n)) = A057163(A154451(A057163(n))). Cf. A069770, A154453.
Differs from A082345 for the first time at n=49, where a(49)=26, while A082345(49)=27. Differs from A122327 for the first time at n=26, where a(26)=49, while A122327(26)=50. Differs from A129612 for the first time at n=195, where a(195)=92, while A129612(195)=91.
Showing 1-5 of 5 results.

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