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Signature permutation of a nonrecursive Catalan automorphism: row 1653002 of table A089840.
4

%I #10 Jun 30 2019 20:22:41

%S 0,1,3,2,6,7,8,5,4,14,15,16,17,18,19,20,21,11,12,22,13,9,10,37,38,39,

%T 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,28,29,59,30,

%U 31,32,60,61,62,33,34,63,35,23,24,64,36,25,26,27,107,108,109,110,111

%N Signature permutation of a nonrecursive Catalan automorphism: row 1653002 of table A089840.

%C It is possible to recursively construct more of these kinds of nonrecursive automorphisms, which by default (if A057515(n) > 1) work as *A074679 and otherwise apply the previous automorphism of this construction process (here *A074679 itself) to the left subtree of a binary tree, before the whole tree is swapped with *A069770. Do the associated cycle-count sequences converge to anything interesting?

%C This automorphism 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...B..............................

%C ............................\./..........\./...............................

%C ..B...C.....A...B........A...x............x...C...A..()...............()..A

%C ...\./.......\./..........\./..............\./.....\./.................\./.

%C A...x....-->..x...C........x..()...-->..()..x.......x..()....-->....()..x..

%C .\./...........\./..........\./..........\./.........\./.............\./...

%C ..x.............x............x............x...........x...............x....

%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 (*A123695! s) (cond ((null? s) s) ((pair? (cdr s)) (*A074679! s)) ((pair? (car s)) (*A074679! (car s)) (*A069770! s))) s)

%Y Inverse: A123696. Row 1653002 of A089840. Variant of A074679.

%K nonn

%O 0,3

%A _Antti Karttunen_, Oct 11 2006