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%I #5 Mar 31 2012 13:21:13
%S 0,1,3,2,8,5,6,4,7,22,13,15,12,14,19,21,16,11,18,10,20,17,9,64,36,41,
%T 35,40,52,53,38,34,39,55,51,37,54,60,63,32,62,31,56,59,47,33,50,27,58,
%U 49,26,43,44,29,61,30,24,57,48,25,46,42,28,23,45,196,106,120,105,119
%N Signature-permutation of a Catalan automorphism: composition of A069772 and A125976.
%C Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.
%H A. Karttunen, <a href="/A126313/b126313.txt">Table of n, a(n) for n = 0..2055</a>
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%Y Inverse: A126314. a(n) = A069772(A125976(n)) = A126290(A069772(n)) = A126315(A057164(n)). The number of cycles, number of fixed points, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127277, A127278, A127279 and A127280. The fixed points are given by A127306. Note the curiosity: this automorphism partitions the A000108(8) = 1430 Catalan structures of size eight (e.g. Dyck paths of length 16) into 79 equivalence classes, of which the largest contains 79 members.
%K nonn
%O 0,3
%A _Antti Karttunen_, Jan 16 2007