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This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ..." and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.

This can be also considered as a "more recursive" variant of A057501 or A057503 or A057161.

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.

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 "FORK". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the both branches (new ones, possibly changed by the given automorphism). I.e. this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures FORK and !FORK 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 A122202.

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 "SPINE". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the new right-hand side branch. The associated Scheme-procedures SPINE and !SPINE 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 A122204.

The recursion scheme SPINE has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. 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 A089840 form a (proper) subset of the rows of this table.

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.

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

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 "NEPEED". In this recursion scheme the algorithm first recurses down to all subtrees, before the given automorphism is applied at the root of general tree. I.e., this corresponds to the post-order (postfix) traversal of a Catalan structure, when it is interpreted as a general tree. The associated Scheme-procedures NEPEED and !NEPEED 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 A122283.

The recursion scheme KROF (described in A122202) is equivalent to a composition of recursion schemes ENIPS (described in A122204) and NEPEED, i.e., KROF(f) = NEPEED(ENIPS(f)) holds for all Catalan automorphisms f. Because of the "universal property of folds", 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 NEPEED(f) = KROF(ENIPS^{-1}(f)). 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 A122284 form a (proper) subset of the rows of table A122202. E.g., row 1 of A122284 is row 15 of A122202, row 2 of A122284 is row 3617 of A122202, row 12 of A122284 is row 65167 of A122202, row 15 of A122284 is row 169 of A122202. - Antti Karttunen, May 25 2007

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

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 "KROF". In this recursion scheme the algorithm first recurses down to the both branches, before the given automorphism is applied at the root of binary tree. I.e., this corresponds to the post-order (postfix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures KROF and !KROF 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 A122201.

The recursion scheme KROF is equivalent to a composition of recursion schemes ENIPS (described in A122204) and NEPEED (described in A122284), i.e., KROF(f) = NEPEED(ENIPS(f)) holds for all Catalan automorphisms f. Because of the "universal property of folds", these recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Specifically, if g = KROF(f), then (f s) = (g (cons (g^{-1} (car s)) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme KROF, we compose g with its own inverse applied to the car- and cdr-branches of a S-expression (i.e., the left and right subtrees in the context of binary trees). This implies that for any nonrecursive automorphism f of the table A089840, KROF^{-1}(f) is also in A089840, which in turn implies that all rows of table A089840 can be found also in table A122202 (e.g., row 1 of A089840 (A069770) occurs here as row 1654720) and furthermore, the table A122290 contains the rows of both tables, A122202 and A089840 as its subsets. Similar notes apply to recursion scheme FORK described in A122201. - Antti Karttunen, May 25 2007

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", or equivalently row n is obtained as ENIPS(SPINE(n-th row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122286. This table contains also all the rows of A122204 and A089840.

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153142. See further comments there and in A153141.

This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(*A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack.

Note: the name in Finnish is "Naks".

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "SPINE", or equivalently row n is obtained as SPINE(ENIPS(n-th row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122285. This table contains also all the rows of A122203 and A089840.