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%I #11 Mar 17 2014 01:53:04
%S 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,
%T 1,0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,
%U 21,9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,17,14,13,12
%N Signature permutations of DEEPEN-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 "DEEPEN". In this recursion scheme the given automorphism is first applied at the root of general tree, before the algorithm recurses down to all subtrees. I.e., this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a general tree. The associated Scheme-procedures DEEPEN and !DEEPEN can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122284.
%C The recursion scheme FORK (described in A122201) is equivalent to a composition of recursion schemes SPINE (described in A122203) and DEEPEN, i.e., FORK(f) = DEEPEN(SPINE(f)) holds for all Catalan automorphisms f. These recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Thus we can equivalently define that DEEPEN(f) = FORK(SPINE^{-1}(f)). Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A122283 form a (proper) subset of the rows of table A122201. E.g., row 1 of A122283 is row 21 of A122201, row 2 of A122283 is row 3613 of A122201, row 17 of A122283 is row 65352 of A122201, row 21 of A122283 is row 251 of A122201. - _Antti Karttunen_, May 25 2007
%D A. Karttunen, paper in preparation, draft available by e-mail.
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o (Scheme:) (define (DEEPEN foo) (letrec ((bar (lambda (s) (map bar (foo s))))) bar))
%o (define (!DEEPEN foo!) (letrec ((bar! (lambda (s) (foo! s) (for-each bar! s) s))) bar!))
%Y The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A122301, 2: A122300, 3: A122303, 4: A122305, 5: A122307, 6: A122309, 7: A122311, 8: A122313, 9: A122315, 10: A122317, 11: A122319, 12: A122321, 13: A122323, 14: A122325, 15: A122327, 16: A122329, 17: A122331, 18: A122333, 19: A122335, 20: A122337, 21: A122339. See also tables A089840, A122200, A122201-A122204, A122285-A122288, A122289-A122290.
%K nonn,tabl
%O 0,4
%A _Antti Karttunen_, Sep 01 2006
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