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%I #3 Mar 31 2012 13:21:16
%S 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,
%T 1,0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,
%U 18,9,8,7,4,4,4,3,2,1,0,12,20,11,12,8,7,5,5,4,3,2,1,0,13,21,14,13,12
%N Signature permutations of INORDER-transformations of non-recursive Catalan automorphisms in table A089840.
%C Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "INORDER". In this recursion scheme the given automorphism is applied at the root of binary tree after the algorithm has recursed down the car-branch (the left hand side tree in the context of binary trees), but before the algorithm recurses down to the cdr-branch (the right hand side of the binary tree, with respect to the new orientation of branches, possibly changed by the applied automorphism). I.e. this corresponds to the depth-first in-order traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures INORDER and !INORDER can be used to obtain such a transformed automorphism from any constructively (or respectively: destructively) implemented automorphism. Each row occurs only once in this table and similar notes as given e.g. for table A122202 apply here, e.g. the rows of A089840 all occur here as well. This transformation has many fixed points besides the trivial identity automorphism *A001477: at least *A069770, *A089863 and *A129604 stay as they are. Inverses of these permutations can be found in table A130401.
%o (Scheme:) (define (INORDER f) (letrec ((g (lambda (s) (cond ((not (pair? s)) s) (else (let ((t (f (cons (g (car s)) (cdr s))))) (cons (car t) (g (cdr t))))))))) g))
%o (define (!INORDER f!) (letrec ((g! (lambda (s) (cond ((pair? s) (g! (car s)) (f! s) (g! (cdr s)))) s))) g!))
%Y Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A073284, 3: A122341, 4: A130381, 5: A130383, 6: A130385, 7: A122350, 8: A082341, 9: A130387, 10: A130389, 11: A130391, 13: A130393, 14: A130395, 15: A130397, 16: A130927, 17: A071657, 18: A130929, 19: A130931, 20: A130933, 21: A089863. Other rows: row 1654694: A073280, row 1654720: A129604.
%Y Cf. also tables A089840, A122201-A122204, A130402-A130403.
%Y Cf. As a sequence differs from A130401 for the first time at n=80, where a(n)=11, while A130401(n)=14.
%K nonn,tabl
%O 0,4
%A _Antti Karttunen_, Jun 11 2007
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