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Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the first element in place if the length (A057515(n)) is odd.
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%I #14 Jan 07 2022 08:27:02

%S 0,1,2,3,4,6,5,7,8,9,11,10,16,19,14,15,12,17,18,13,20,21,22,23,25,24,

%T 30,33,37,29,26,44,47,27,53,56,60,28,39,38,43,52,42,40,31,45,46,32,48,

%U 49,50,51,41,34,54,55,35,57,58,59,36,61,62,63,64,65,67,66,72,75,79,71

%N Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the first element in place if the length (A057515(n)) is odd.

%C This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 5 ... 2n 2n+1), permutes it as (1 3 2 5 4 ... 2n+1 2n).

%H Antti Karttunen, <a href="/A130373/b130373.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>

%F a(n) = A057508(A130374(A057508(n))) = A057164(A130374(A057164(n))).

%Y Cf. A057508, A057164, A057164.

%Y SPINE and ENIPS transform of *A130340 (transformations explained in A122203 and A122204).

%Y The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.

%K nonn

%O 0,3

%A _Antti Karttunen_, Jun 05 2007