A071661 -id:A071661 - cp's OEIS frontend

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

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.

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)).

A126312 Fixed points of permutation A071661/A071662.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 22, 23, 30, 55, 64, 65, 98, 158, 196, 197, 318, 484, 625, 626, 687, 1042, 1549, 1973, 2055, 2056, 2376, 3471, 5113, 6558, 6917, 6918, 8191, 11763, 17268, 22277, 23713, 23714, 24331, 28360, 40491, 59362, 76942, 81754, 82499

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Those i for which A071661(i)=i, i.e. for which A057163(A057164(i)) = A057164(A057163(i)). These appear to consist of just those general plane trees which are symmetric and will stay symmetric also after the underlying plane binary tree has been reflected, i.e. for which A057164(i)=i and A057164(A057163(i)) = A057163(i). See comments at A123050 and A080070. The sequence seems to give also the fixed points of the permutation A125977/A125978.

D. Callan, A Bijection on Dyck Paths and Its Cycle Structure, 2006, 17pp.

A079437 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Original entry on oeis.org

1, 1, 2, 3, 6, 16, 36, 83, 190, 448, 1056, 2514, 5872, 13806, 32424, 76609, 181434, 432062, 1032716

Offset: 0

Antti Karttunen, Jan 27 2003

That is, number of orbits to which "Catalan bijections" A071661/A071662 partition each A000108(n) Catalan tree structures encoded in A014486[A014137(n-1)..A014138(n-1)].

Antti Karttunen, C-program for counting the initial terms of this sequence

Occurs in A073201 as row 13373289.

Cf. A079438, A079439, A057507, A079441.

A079439 Longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 12, 36, 72, 147, 294, 336, 1068, 5076, 5760, 14742, 58968, 135288, 328176

Offset: 0

Antti Karttunen, Jan 27 2003

Antti Karttunen, C-program for counting the initial terms of this sequence

Occurs in A073203 as row 13373289. Cf. A079437, A079438, A079443.

A089403 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A071661/A071662.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 60, 360, 7560, 582120, 7567560, 141965858433999600

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

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

Cf. A089871, A089875.

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))).

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.

A071667 Permutation A057505 applied five times ("^5").

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 11, 17, 12, 10, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 39, 54, 34, 38, 37, 63, 56, 57, 40, 42, 59, 43, 50, 36, 33, 47, 48, 29, 28, 58, 44, 49, 35, 30, 46, 32, 27, 25, 45, 31, 26, 24, 23, 196, 191, 192, 178, 177

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: A071668 and also its car/cdr-flipped conjugate, i.e. A071667(n) = A057163(A071668(A057163(n))) = A057505(A071665(n)). Cf. also A071661, A071663, A071669.

A071662 Permutation A057506 applied twice ("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, 43, 56, 63, 24, 46, 32, 47, 59, 25, 48, 28, 37, 54, 34, 40, 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)
Indranil Ghosh, Python Program for computing this sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071661 and also its car/cdr-flipped conjugate, i.e. A071662(n) = A057163(A071661(A057163(n))) = A057506(A057506(n)). Cf. also A071664, A071666, A071668, A071670.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A126312 Fixed points of permutation A071661/A071662.

Views

Author

Keywords

Comments

Links

A079437 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Views

Author

Keywords

Comments

Links

Crossrefs

A079439 Longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Views

Author

Keywords

Links

Crossrefs

A089403 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A071661/A071662.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Links

Crossrefs

A071667 Permutation A057505 applied five times ("^5").

Views

Author

Keywords

Links

Crossrefs

A071662 Permutation A057506 applied twice ("squared").

Views

Author

Keywords

Links

Crossrefs