A125977 -id:A125977 - cp's OEIS frontend

A126317 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

Original entry on oeis.org

1, 1, 2, 3, 6, 16, 36, 79, 162, 316, 604, 1204, 2244

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

A126318 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 11, 45, 257, 575, 2470, 10892, 30297

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

A126319 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 33, 9061920, 1252445414220, 126032376305404800, 50110448042127911907268800, 13399946812028296616282674883512406948355335893125182077721607466200299913000

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

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))).

A125976 Signature-permutation of Kreweras' 1970 involution on Dyck paths.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

Lalanne shows in the 1992 paper that this automorphism preserves the sum of peak heights, i.e., that A126302(a(n)) = A126302(n) for all n. Furthermore, he also shows that A126306(a(n)) = A057514(n)-1 and likewise, that A057514(a(n)) = A126306(n)+1, for all n >= 1.
Like A069772, this involution keeps symmetric Dyck paths symmetric, but not necessarily same.
The number of cycles and fixed points in range [A014137(n-1)..A014138(n-1)] of this involution seem to be given by A007595 and the "aerated" Catalan numbers [1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, ...], thus this is probably a conjugate of A069770 (as well as of A057163).

A. Karttunen, Table of n, a(n) for n = 0..2055
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
J.-C. Lalanne, Une Involution sur les Chemins de Dyck, European J. Combin. 13 (1992), no. 6, 477-487.
Index entries for signature-permutations of Catalan automorphisms

Cf. A080300, A125974, A014486.
Cf. A057163, A069770, A007595, A014137, A014138.
Compositions and conjugations with other automorphisms: A125977-A125979, A125980, A126290.

a(n) = A080300(A125974(A014486(n))).

A125978 Signature-permutation of a Catalan automorphism: composition of A125976 and A057163.

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, 48, 56, 63, 24, 46, 32, 43, 59, 25, 47, 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, Jan 02 2007

nonn

Index entries for signature-permutations of Catalan automorphisms

Inverse: A125977. a(n) = A125976(A057163(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.

A125980 Signature-permutation of a Catalan automorphism: A057163-conjugate of A125976.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

This is self-inverse permutation (involution) of the nonnegative integers.

Index entries for signature-permutations of Catalan automorphisms

a(n) = A057163(A125976(A057163(n))) = A125977(A057163(n)) = A057163(A125978(n)).

cp's OEIS Frontend

A126317 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

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A126318 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

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A126319 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A125977/A125978.

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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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A125976 Signature-permutation of Kreweras' 1970 involution on Dyck paths.

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A125978 Signature-permutation of a Catalan automorphism: composition of A125976 and A057163.

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A126312 Fixed points of permutation A071661/A071662.

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A125980 Signature-permutation of a Catalan automorphism: A057163-conjugate of A125976.

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