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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 "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.

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".

The signature-permutation of the Catalan automorphism which is derived from the third non-recursive Catalan automorphism *A089850 with recursion schema ENIPS (see A122204 for the definition). This automorphism is also produced when automorphism *A069768 is applied to the right-hand side subtree of the given binary tree, with the left side left intact.

The signature-permutation of the self-inverse Catalan automorphism which is derived from automorphism *A130341 with recursion scheme FORK or KROF.

Derived from automorphism *A130341 with recursion scheme ENIPS.