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A122204
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Signature permutations of ENIPS-transformations of non-recursive Catalan automorphisms in table A089840.
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54
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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, 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, 21, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 14, 13, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 17, 10, 12, 13
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OFFSET
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0,4
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COMMENTS
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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 "ENIPS". In this recursion scheme the algorithm first recurses down to the right-hand side branch of the binary tree, before the given automorphism is applied at its root. This corresponds to the fold-right operation applied to the Catalan structure, interpreted e.g. as a parenthesization or a Lisp-like list, where (lambda (x y) (f (cons x y))) is the binary function given to fold, with 'f' being the given automorphism. The associated Scheme-procedures ENIPS and !ENIPS 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 A122203.
Because of the "universal property of folds", the recursion scheme ENIPS has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = ENIPS(f), then (f s) = (g (cons (car s) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme ENIPS, we compose g with its own inverse applied to the cdr-branch of a S-expression (i.e. the right subtree in the context of binary trees). This implies that for any non-recursive automorphism f in the table A089840, ENIPS^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.
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REFERENCES
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A. Karttunen, paper in preparation, draft available by e-mail.
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LINKS
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PROG
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(MIT Scheme:) (define (ENIPS foo) (lambda (s) (fold-right (lambda (x y) (foo (cons x y))) '() s)))
(define (!ENIPS foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (bar! (cdr s)) (foo! s))) s))) bar!))
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CROSSREFS
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Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069768, 2: A057510, 3: A130342, 4: A130348, 5: A130346, 6: A130344, 7: A122282, 8: A082340, 9: A130354, 10: A130352, 11: A130350, 12: A057502, 13: A130364, 14: A130366, 15: A069770, 16: A130368, 17: A074686, 18: A130356, 19: A130358, 20: A130362, 21: A130360. Other rows: row 169: A089859, row 253: A123718, row 3608: A129608, row 3613: A072796, row 65167: A074679, row 79361: A123716.
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KEYWORD
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AUTHOR
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STATUS
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approved
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