|
%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
|
