A071667 -id:A071667 - cp's OEIS frontend

A089877 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

Original entry on oeis.org

1, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 10, 0, 5, 0, 10, 0, 45, 0, 10, 0

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

The number of n-node binary trees fixed by the corresponding automorphism(s).

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

A089876 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

Original entry on oeis.org

1, 1, 1, 2, 3, 14, 22, 62, 127, 320, 756, 1888, 4440, 10996, 26784, 67004, 166449, 417848, 1041262, 2596506, 6427116

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

A089878 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 24, 72, 144, 147, 588, 672, 2136, 10152, 11496, 29484, 117936, 270576, 656352, 2062368, 3184728

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

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

For the terms a(0)-a(20) differs from A057545 only at n=14 where a(14)=11496 != A057545(14)=11520 and at n=20, where a(20)=3184728 while A057545(20)=4040160.

A089879 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

Original entry on oeis.org

1, 1, 2, 6, 6, 6, 24, 144, 3024, 232848, 3027024, 56786343373599840

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

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

A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 15, 19, 16, 10, 11, 14, 9, 64, 63, 59, 62, 58, 50, 49, 55, 61, 57, 46, 48, 54, 45, 36, 35, 32, 34, 31, 41, 40, 52, 60, 56, 43, 47, 53, 44, 27, 26, 29, 33, 30, 38, 39, 51, 42, 24, 25, 28, 37, 23, 196, 195, 190, 194, 189

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ..." and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.

This can be also considered as a "more recursive" variant of A057501 or A057503 or A057161.

D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006).

Antti Karttunen, Table of n, a(n) for n = 0..2055
Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Robert Donaghey and Louis W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, Vol. 23, No. 3 (1977), pp. 291-301.
Indranil Ghosh, Python program for computing this sequence (after the functions mentioned in the OEIS wiki).
Antti Karttunen, Illustration of symmetric general trees whose A057163-reflection is also symmetric (ones that occur in 2-cycles of A057505/A057506)
Antti Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki regarding the 2-cycles of this automorphism)
D. E. Knuth, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057506.
The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
Other related permutations: A057501, A057503, A057161.
Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
Row 17 of table A122288.
Cf. A080981 (the "primitive elements" of this automorphism), A079438, A079440, A079442, A079444, A080967, A080968, A080972, A080272, A080292, A083929, A080973, A081164, A123050, A125977, A126312.

map(CatalanRankGlobal,map(DonagheysM, A014486)); or map(CatalanRankGlobal,map(DeepRotateTriangularization, A014486));
DonagheysM := n -> pars2binexp(DonagheysMP(binexp2pars(n)));
DonagheysMP := h -> `if`((0 = nops(h)),h,[op(DonagheysMP(car(h))),DonagheysMP(cdr(h))]);
DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(a(A072772(n)))). [This recurrence reflects the S-expression implementation given first in the Program section: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some languages), and A057548 corresponds to unary form of function 'list'].
As a composition of related permutations:
a(n) = A057164(A057163(n)).
a(n) = A057163(A057506(A057163(n))).

A071661 Permutation A057505 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 11, 19, 17, 20, 12, 10, 15, 21, 18, 13, 22, 23, 37, 42, 28, 51, 44, 53, 30, 25, 39, 56, 47, 33, 60, 45, 54, 57, 48, 61, 31, 34, 26, 24, 38, 40, 43, 29, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 107, 121, 79, 149, 126

Offset: 0

Antti Karttunen, May 30 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071662 and also its car/cdr-flipped conjugate, i.e. A071661(n) = A057163(A071662(A057163(n))) = A057505(A057505(n)). Cf. also A071663, A071665, A071667, A071669.

A071663 Permutation A057505 applied three times ("cubed").

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2002

nonn

Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071664 and also its car/cdr-flipped conjugate, i.e. A071663(n) = A057163(A071664(A057163(n))) = A057505(A071661(n)). Cf. also A071665, A071667, A071669.

A071665 Permutation A057505 applied four times ("^4"), permutation A071661 squared.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 12, 16, 21, 10, 18, 11, 14, 20, 13, 15, 19, 22, 23, 45, 31, 44, 58, 26, 49, 30, 42, 57, 35, 40, 56, 63, 24, 46, 32, 43, 59, 25, 47, 28, 37, 54, 48, 34, 53, 62, 27, 50, 29, 38, 55, 33, 39, 51, 61, 36, 41, 52, 60, 64, 65, 129, 87, 128, 170, 73

Offset: 0

Antti Karttunen, May 30 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071666 and also its car/cdr-flipped conjugate, i.e. A071665(n) = A057163(A071666(A057163(n))) = A057505(A071663(n)) = A071661(A071661(n)). Cf. also A071667, A071669.

A071668 Permutation A057506 applied five times ("^5").

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 15, 19, 16, 10, 11, 14, 9, 64, 63, 59, 62, 58, 50, 49, 55, 61, 57, 46, 34, 54, 45, 36, 35, 32, 40, 31, 41, 43, 52, 60, 56, 47, 48, 53, 44, 27, 26, 29, 33, 30, 38, 39, 51, 42, 24, 25, 28, 37, 23, 196, 195, 190, 194, 189

Offset: 0

Antti Karttunen, May 30 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071667 and also its car/cdr-flipped conjugate, i.e. A071668(n) = A057163(A071667(A057163(n))) = A057506(A071666(n)). Cf. also A071662, A071664, A071670.

A071669 Permutation A057505 applied six times, permutation A071661 cubed, permutation A071663 squared.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2002

nonn

Inverse permutation: A071670 and also its car/cdr-flipped conjugate. See formulas.

Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, On the fixed points of A071661, OEIS Wiki, notes concerning this and related sequences. See especially at the end.
Antti Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence).
Index entries for sequences that are permutations of the natural numbers.

Cf. A057163, A057505, A071661, A071663, A071665, A071667, A071669, A071670.

a(n) = A057163(A071670(A057163(n))) = A057505(A071667(n)) = A071661(A071661(A071661(n))) = A071663(A071663(n)).

cp's OEIS Frontend

A089877 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

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A089876 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

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A089878 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

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A089879 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A071667/A071668.

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A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

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Maple

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A071661 Permutation A057505 applied twice ("squared").

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A071663 Permutation A057505 applied three times ("cubed").

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A071665 Permutation A057505 applied four times ("^4"), permutation A071661 squared.

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A071668 Permutation A057506 applied five times ("^5").

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A071669 Permutation A057505 applied six times, permutation A071661 cubed, permutation A071663 squared.

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