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%I #8 Jan 13 2013 06:14:42
%S 0,1,2,3,7,6,5,4,8,9,10,21,20,19,14,15,18,17,16,13,12,11,22,45,46,44,
%T 42,43,31,32,30,28,29,63,62,61,60,54,55,53,51,52,26,27,25,23,24,59,58,
%U 57,56,40,41,39,37,38,50,49,48,47,36,35,34,33,64,65,67,66,68,69,170
%N Self-inverse permutation of natural numbers induced by the automorphism xReflectHandshakes acting on the parenthesizations encoded by A014486.
%C This automorphism reflects over the x-axis the interpretation n (the non-crossing handshakes) of Stanley's exercise 19.
%C Note that DeepRev (A057164) reflects over y-axis.
%C This transformation keeps palindromic parenthesizations/Dyck paths/rooted planar trees palindromic, but not necessarily same, meaning that this induces a permutation on the sequence A061855 (= A069766).
%H A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</a> (Includes the complete Scheme program for computing this sequence)
%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catalan.pdf">Exercises on Catalan and Related Numbers</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%o (Scheme function implementing this automorphism on list-structures:) (define (xReflectHandshakes a) (DeepRev (RotateHandshakes180 a)))
%o (define (DeepRev lista) (cond ((not (pair? lista)) lista) ((null? (cdr lista)) (cons (DeepRev (car lista)) (list))) (else (append (DeepRev (cdr lista)) (DeepRev (cons (car lista) (list)))))))
%Y Composition of A057164 and A069771 in either order, i.e. A069772(n) = A057164(A069771(n)) = A069771(A057164(n)). Cf. also A061855, A069766, A057501, A069888, A069889.
%K nonn
%O 0,3
%A Antti Karttunen, Apr 16 2002
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