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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.

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 "RIBS".

In this recursion scheme the given automorphism is applied to all (toplevel) subtrees of the Catalan structure, when it is interpreted as a general tree. Permutations in this table form a countable group, which is isomorphic with the group in A089840. (The RIBS transformation gives the group isomorphism.)

Furthermore, row n of this table is also found as the row A123694(n) in tables A122203 and A122204. If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution) and the count of fixed points of the automorphism A122200[n] by INVERT(RIGHT(f)).

The associated Scheme-procedures RIBS and !RIBS can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism.

This sequence agrees with A025581 in its initial terms, but then diverges from it. - Antti Karttunen, May 11 2008

This automorphism 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...........C...B

....\./.............\./

.A...x....-->....A...x.................A..().........A...()..

..\./.............\./...................\./....-->....\./...

...x...............x.....................x.............x....

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

In terms of S-expressions, this automorphism swaps cadr and cddr of an S-exp if its length > 1.

Look at the example in A069770 to see how this will produce the given sequence of integers.

Recursive transformation ENIPS for Catalan bijections has a well-defined inverse (see the definition & comments at A122204). For all Catalan bijections in A089840 that inverse produces a bijection which is itself in A089840. This sequence gives the indices to those positions where each ("primitive", non-recursive bijection) of A089840(n) occurs "atavistically" amongst the more complex recursive bijections in A122204. I.e. A122204(a(n)) = A089840(n). Similarly, other "atavistic forms" resurface as: A122287(a(n)) = A122201(n), A122286(a(n)) = A122203(n) and A122202(a(n)) = A122284(n). See also comments at A153833.

There exists similar atavistic index sequences computed for FORK (A122201) and KROF (A122202). Both start as 0,1654720,... (see A129604). This implies that regardless of how complex recursive derivations from A089840 one forms by repeatedly applying SPINE, ENIPS, FORK and/or KROF in some order (finite number of times), all the original primitive non-recursive elements of A089840 will eventually appear at some positions.

Other known terms: a(12)=65167, a(13)=65178, a(14)=65236, a(15)=169, a(16)=65302, a(22)-a(44) = 1656351, 1656576, 1656777, 1656628, 1656704, 1659507, 1659538, 1659653, 1659798, 1659685, 1659830, 1660155, 1660582, 1660439, 1660476, 1660621, 1660196, 1661073, 1660930, 1660859, 1661004, 1661287, 1661360.

Recursive transformation SPINE for Catalan bijections has a well-defined inverse (see the definition & comments at A122203). For all Catalan bijections in A089840 that inverse produces a bijection which is itself in A089840. This sequence gives the indices to those positions where each ("primitive", non-recursive bijection) of A089840(n) occurs "atavistically" amongst the more complex recursive bijections in A122203. I.e. A122203(a(n)) = A089840(n). Similarly, other "atavistic forms" resurface as: A122288(a(n)) = A122202(n), A122285(a(n)) = A122204(n) and A122201(a(n)) = A122283(n). See also comments at A153832.

Other known terms: a(17)-a(44): 65352, 65359, 65604, 65739, 251, 1656303, 1656426, 1656552, 1656628, 1656479, 1661655, 1661816, 1666720, 1684006, 1684221, 1667042, 1667007, 1684152, 1661799, 1661676, 1666759, 1684081, 1684437, 1667151, 1684509, 1667187, 1661961, 1661944.

SPINE-transformation is explained in A122203. This automorphism permutes the top-level of a list of even length (1 ... 2n) as (2n 1 2n-1 2 2n-3 3 ... n+1 n) and when applied to a list of odd length (1 .. 2n+1), permutes it as (2n+1 1 2n 2 2n-1 3 ... n n+1). Used to construct A127287 and A127289.

This self-inverse automorphism is obtained as either SPINE(*A129607) or ENIPS(*A129607). See the definitions given in A122203 and A122204.

Automorphism *A089863 = SPINE(*A129612). See the definition given in A122203.

This self-inverse automorphism is obtained as either SPINE(*A129608) or ENIPS(*A129608). See the definitions given in A122203 and A122204.

The signature-permutation of the Catalan automorphism which is derived from *A069776 with recursion schema SPINE (see A122203 for the definition).