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A127379 Signature-permutation of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

%I A127379 #9 Apr 19 2013 13:52:38
%S A127379 0,1,2,3,4,5,6,8,7,9,10,11,13,12,14,15,19,22,21,16,20,17,18,23,24,25,
%T A127379 27,26,28,29,33,36,35,30,34,31,32,37,38,39,41,40,51,52,60,64,63,56,62,
%U A127379 58,59,42,43,53,54,55,44,61,45,46,47,57,48,50,49,65,66,67,69,68,70,71
%N A127379 Signature-permutation of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).
%C A127379 It's much easier to implement Callan's 2006 bijection for S-expressions if one considers a mirror-image of the graphical description given by Callan (on page 3). Then this automorphism is just RIBS-transformation (explained in A122200) of the automorphism A127377 and Callan's original variant A127381 is obtained as A057164(a(A057164(n))).
%H A127379 A. Karttunen, <a href="/A127379/b127379.txt">Table of n, a(n) for n = 0..2055</a>
%H A127379 D. Callan, <a href="http://www.stat.wisc.edu/~callan/papers/Dyck_bijection/">A Bijection on Dyck Paths and Its Cycle Structure</a>, 2006, 17pp.
%H A127379 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o A127379 (Scheme implementation of this automorphism, destructive, that acts on S-expressions, i.e. list-structures:)
%o A127379 (define (*A127379! s) (for-each *A127377! s) s)
%Y A127379 Inverse: A127380. a(n) = A057164(A127381(A057164(n))). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127384 and A086625 shifted once right. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, ... A127302(a(n)) = A127302(n) holds for all n. A127388 shows a variant which is an involution.
%Y A127379 Differs from A073289 and A122349 for the first time at n=54, where a(n)=54, while A073289(54) = A122349(54) = 61.
%K A127379 nonn
%O A127379 0,3
%A A127379 _Antti Karttunen_, Jan 16 2007