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%I #6 Mar 31 2012 13:21:14
%S 0,1,2,3,4,6,5,7,8,9,11,14,16,19,10,15,12,17,18,13,20,21,22,23,24,28,
%T 30,33,37,39,42,44,47,51,53,56,60,25,29,38,43,52,26,40,31,45,46,32,48,
%U 49,50,27,41,34,54,55,35,57,58,59,36,61,62,63,64,65,66,67,68,69,79,80
%N Signature-permutation of a Catalan automorphism, row 3613 of A089840.
%C This involution effects the following transformation on the binary trees (labels A,B,C,D refer to arbitrary subtrees located on those nodes and () stands for a terminal node.)
%C .....C...D.........A...D
%C ......\./...........\./
%C ...B...X2........C...Y2......B..().......A..()
%C ....\./...........\./.........\./.........\./
%C .A...X1....-->.B...Y1......A...X1..-->.B...Y1
%C ..\./...........\./.........\./.........\./
%C ...X0............Y0..........X0..........Y0
%C Note that automorphism *A072796 = SPINE(*A129605). See the definition given in A122203.
%H A. Karttunen, <a href="/A129605/b129605.txt">Table of n, a(n) for n = 0..2055</a>
%H A. Karttunen, <a href="/A089840/a089840p.txt">Prolog-program which illustrates the construction of this and similar nonrecursive Catalan automorphisms.</a>
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o (Constructive and destructive Scheme implementation of this automorphism. These act on S-expressions, i.e. list-structures:)
%o (define (*A129605 s) (cond ((> (length s) 2) (cons (cadr s) (cons (caddr s) (cons (car s) (cdddr s))))) (else (*A072796 s))))
%o (define (*A129605! s) (cond ((< (length s) 3) (*A072796! s)) (else (let ((org_car (car s))) (set-car! s (cadr s)) (set-car! (cdr s) (caddr s)) (set-car! (cddr s) org_car) s))))
%Y Inverse: A129606.
%K nonn
%O 0,3
%A _Antti Karttunen_, May 22 2007
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