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%I #6 Mar 31 2012 13:21:12
%S 0,1,2,3,4,6,5,8,7,9,10,14,16,19,11,15,12,21,22,13,20,17,18,23,24,25,
%T 26,27,37,38,42,44,47,51,53,56,60,28,29,39,43,52,30,40,31,58,59,32,62,
%U 63,64,33,41,34,57,61,35,54,45,46,36,55,48,49,50,65,66,67,68,69,70,71
%N An involution of nonnegative integers: signature permutation of a nonrecursive Catalan automorphism, row 253 of table A089840.
%C This automorphism either swaps (if A057515(n) > 1) the first two toplevel elements (of a general plane tree, like *A072796 does) and otherwise (if n > 1, A057515(n)=1) swaps the sides of the left hand side subtree of the S-expression (when viewed as a binary tree, like *A089854 does). This is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.
%C ...B...C.............A...C............A...B...........B...A
%C ....\./...............\./..............\./.............\./
%C .A...x.....-->.....B...x................x..()....-->....x..()
%C ..\./...............\./..................\./.............\./
%C ...x....(A072796)....x....................x...(A089854)...x
%C (a . (b . c)) --> (b . (a . c)) / ((a . b) . ()) --> ((b . a) . ())
%C This is the first multiclause automorphism in table A089840 which cannot be represented as a composition of two smaller nonrecursive automorphisms, the property which is also shared by *A123499 and *A123500.
%H A. Karttunen, <a href="/A089840/a089840p.txt">Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.</a>
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123503! s) (cond ((null? s) s) ((pair? (cdr s)) (*A072796! s)) (else (*A089854! s))) s)
%Y Row 253 of A089840. Used to construct A123717 and A123718.
%K nonn
%O 0,3
%A _Antti Karttunen_, Oct 11 2006
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