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%I #15 Jan 27 2014 16:22:51
%S 0,1,2,3,4,5,6,8,7,9,10,11,13,12,14,15,19,22,18,16,21,17,20,23,24,25,
%T 27,26,28,29,33,36,32,30,35,31,34,37,38,39,41,40,51,52,60,64,46,47,59,
%U 50,49,42,43,56,63,55,44,58,45,48,53,62,54,61,57,65,66,67,69,68,70,71
%N Signature-permutation of Callan's 2006 bijection on Dyck Paths.
%C This automorphism is much easier to implement for S-expressions when viewed as a A057164-conjugate of A127379. See the comments in the latter entry.
%H A. Karttunen, <a href="/A127381/b127381.txt">Table of n, a(n) for n = 0..2055</a>
%H D. Callan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1r28">A Bijection on Dyck Paths and its Cycle Structure</a>, The Electronic Journal of Combinatorics 14 (2007), #R28
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%Y Inverse: A127382. a(n) = A057164(A127379(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, ...
%K nonn,look
%O 0,3
%A _Antti Karttunen_, Jan 16 2007