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Shifted once right (beginning as 1,1,1,2,5,9,...) this is maximum cycle size (in the same range) of permutations A085169/A085170, shifted twice right (beginning as 1,1,1,1,2,5,9,...) this is the maximum cycle size in permutations A089867/A089868 and A089869/A089870.

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.

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.......A...B

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

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

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

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

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

That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

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 "FORK", or equivalently row n is obtained as FORK(ENIPS(n-th row of A089840)). See A122201 and A122204 for the description of FORK and ENIPS. Moreover, each row of A122287 can be obtained as the "DEEPEN" transform of the corresponding row in A122286. (See A122283 for the description of DEEPEN). Each row occurs only once in this table. Inverses of these permutations can be found in table A122288. This table contains also all the rows of A122201 and A089840.

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.

This is a rare example of a simply defined Catalan Automorphism where the cycle count sequence (A089411) is not monotone. (See A127296 for a much more complex example.)