login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
OFFSET
0,3
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
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, 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.
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))).
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;
PROG
(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.
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.
Sequence in context: A130362 A085173 A071668 * A122357 A122298 A122337
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 03 2000
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)