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

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

Author

Antti Karttunen, Sep 03 2000

Keywords

• nonn

Comments

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.

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Maple

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;

Formula

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))).