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This is the simplest possible Catalan automorphism after the identity bijection (A001477). It effects the following transformation on the unlabeled rooted plane binary trees (letters A and B refer to arbitrary subtrees located on those vectices):

A B B A

\ / --> \ /

x x

(a . b) -----> (b . a)

Applying this permutation recursively to the right hand side branch of the binary trees produces permutations A069767 and A069768 (that occur at the same index 1 in tables A122203 and A122204), and applying this recursively to the both branches of binary trees (as in pre- or postorder traversal) produces A057163 (which occurs at the same index 1 in tables A122201 and A122202) that reflects the whole binary tree.

For this permutation, A127302(a(n)) = A127302(n) for all n, [or equally, A153835(a(n)) = A153835(n)], and likewise for all such recursive derivations as mentioned above.

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

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.

This bijection effects the following transformation on the unlabeled rooted plane general trees (letters A, B, C, etc. refer to arbitrary subtrees located on those vertices):

A A A B B A A B C B A C

| --> | \ / --> \ / \ | / --> \ | /

| | \./ \./ \|/ \|/ etc.

I.e., it keeps "planted" (root degree = 1) trees intact, and swaps the two leftmost toplevel subtrees of the general trees that have a root degree > 1.

On the level of underlying binary trees that general trees map to (see, e.g., 1967 paper by N. G. De Bruijn and B. J. M. Morselt, or consider lists vs. dotted pairs in Lisp programming language), this bijection effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).

B C A C

\ / \ /

A x --> B x A () A ()

\ / \ / \ / --> \ /

x x x x

(a . (b . c)) -> (b . (a . c)) (a . ()) ---> (a . ())

Note that the first clause corresponds to what is called "generator pi_0" in Thompson's group V. (See also A074679, A089851 and A154121 for other related generators.)

Look at the example section to see how this will produce the given sequence of integers.

Applying this permutation recursively down the right hand side branch of the binary trees (or equivalently, along the topmost level of the general trees) produces permutations A057509 and A057510 (that occur at the same index 2 in tables A122203 and A122204) that effect "shallow rotation" on general trees and parenthesizations. Applying it recursively down the both branches of binary trees (as in pre- or postorder traversal) produces A057511 and A057512 (that occur at the same index 2 in tables A122201 and A122201) that effect "deep rotation" on general trees and parenthesizations.

For this permutation, A127301(a(n)) = A127301(n) for all n, which in turn implies A129593(a(n)) = A129593(n) for all n, likewise for all such recursively generated bijections as A057509 - A057512. Compare also to A072797.

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

This is a permutation of natural numbers induced when "noncrossing handshakes", i.e., Stanley's interpretation (n), "n nonintersecting chords joining 2n points on the circumference of a circle", are rotated.

The same permutation is induced when the root position of plane trees (Stanley's interpretation (e)) is successively changed around the vertices.

For a good illustration how the rotation of the root vertex works, please see the Figure 6, "Rotation of an ordered rooted tree" in Torsten Mütze's paper (on page 24 in 20 May 2014 revision).

For yet another application of this permutation, please see the attached notes for A085197.

By "recursivizing" either the left or right hand side argument of A085201 in the formula, one ends either with A057161 or A057503. By "recursivizing" the both sides, one ends with A057505. - Antti Karttunen, Jun 06 2014

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

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right 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 A153141. See further comments there.

This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) 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 a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.

Note: the name in Finnish is "Niks".

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122204 with the recursion scheme "SPINE", i.e. row n is obtained as SPINE(A057163 o ENIPS(A089840[n]) o A057163). 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 A130402. This table contains also all the rows of A122203 and A089840.