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A130373 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.

%I A130373 #14 Jan 07 2022 08:27:02
%S A130373 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 A130373 30,33,37,29,26,44,47,27,53,56,60,28,39,38,43,52,42,40,31,45,46,32,48,
%U A130373 49,50,51,41,34,54,55,35,57,58,59,36,61,62,63,64,65,67,66,72,75,79,71
%N A130373 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 A130373 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 A130373 Antti Karttunen, <a href="/A130373/b130373.txt">Table of n, a(n) for n = 0..2055</a>
%H A130373 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%F A130373 a(n) = A057508(A130374(A057508(n))) = A057164(A130374(A057164(n))).
%Y A130373 Cf. A057508, A057164, A057164.
%Y A130373 SPINE and ENIPS transform of *A130340 (transformations explained in A122203 and A122204).
%Y A130373 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 A130373 nonn
%O A130373 0,3
%A A130373 _Antti Karttunen_, Jun 05 2007