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A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486. 55
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 (list; graph; refs; listen; history; text; internal format)



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.


R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.

R. Donaghey & L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.

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

A. Karttunen, Illustration of symmetric general trees whose A057163-reflection is also symmetric (ones that occur in 2-cycles of A057505/A057506)

A. 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,

Indranil Ghosh, Python program for computing this sequence (after the functions mentioned in the OEIS wiki)

Index entries for signature-permutations of Catalan automorphisms


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))).


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;


(Scheme functions implementing this automorphism on S-expressions, three different variants):

(define (*A057505 a) (cond ((null? a) a) (else (append (*A057505 (car a)) (list (*A057505 (cdr a)))))))

(define (*A057505 bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (car lt) (cons (*A057505 (cdr lt)) nt))))))

(define (*A057505! s) (cond ((pair? s) (*A057505! (car s)) (*A057505! (cdr s)) (*A057501! s))) s)

;; A version working directly on nonnegative integers (definec is a memoization macro from Antti Karttunen's IntSeq-library):

(definec (A057505 n) (if (zero? n) n (A085201bi (A057505 (A072771 n)) (A057548 (A057505 (A072772 n)))))) ;; A085201bi, see: A085201.


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.

Sequence in context: A130362 A085173 A071668 * A122357 A122298 A122337

Adjacent sequences:  A057502 A057503 A057504 * A057506 A057507 A057508




Antti Karttunen, Sep 03 2000



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Last modified January 17 17:23 EST 2019. Contains 319250 sequences. (Running on oeis4.)