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A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

%I A130374 #14 Jan 07 2022 07:23:37
%S A130374 0,1,2,3,4,6,5,7,8,9,10,14,16,19,11,15,12,17,18,13,20,21,22,23,25,24,
%T A130374 26,27,37,38,42,44,47,51,53,56,60,28,29,39,43,52,30,40,31,45,46,32,48,
%U A130374 49,50,33,41,34,54,55,35,57,58,59,36,61,62,63,64,65,66,70,72,75,67,71
%N A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.
%C A130374 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 ... 2n-1 2n 2n+1), permutes it as (2 1 4 3 ... 2n 2n-1 2n+1).
%H A130374 Antti Karttunen, <a href="/A130374/b130374.txt">Table of n, a(n) for n = 0..2055</a>
%H A130374 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o A130374 (Destructive Scheme implementation of this automorphism, which acts on S-expressions, i.e., list-structures:)
%o A130374 (define (*A130374! s) (cond ((pair? s) (*A072796! s) (if (pair? (cdr s)) (*A130374! (cddr s))))) s)
%Y A130374 Cf. a(n) = A057508(A130373(A057508(n))) = A057164(A130373(A057164(n))) = A127285(A127288(n)) = A127287(A127286(n)). Also a(A085223(n)) = A130370(A122282(A130369(A085223(n)))) holds for all n>=0. 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 A130374 nonn,look
%O A130374 0,3
%A A130374 _Antti Karttunen_, Jun 05 2007