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Search: id:A057505
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| A057505 |
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Permutation of natural numbers induced by the automorphism DonagheysM (or DeepRotateTriangularization) acting on the parenthesizations encoded by A014486. |
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+0
48
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| 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; listen)
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OFFSET
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0,3
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COMMENT
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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 "deeply recursive" variant of A057501 or a recursive variant of A057161.
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REFERENCES
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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.
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LINKS
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A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
Index entries for sequences that are permutations of the natural numbers
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MAPLE
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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;
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PROGRAM
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(Two alternative Scheme functions implementing this automorphism on list-structures:)
(define (DonagheysM a) (cond ((null? a) a) (else (append (DonagheysM (car a)) (list (DonagheysM (cdr a)))))))
(define (DeepRotateTriangularization bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (car lt) (cons (DeepRotateTriangularization (cdr lt)) nt))))))
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CROSSREFS
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Inverse permutation: A057506 and also its car/cdr-flipped conjugate, i.e. A0057505(n) = A057163(A057506(A057163(n))). Composition of A057163 & A057164, i.e. A057505(n) = A057164(A057163(n)).
The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
Row 17 of table A122288.
Sequence in context: A130362 A085173 A071668 this_sequence A122357 A122298 A122337
Adjacent sequences: A057502 A057503 A057504 this_sequence A057506 A057507 A057508
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen Sep 03 2000
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