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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 "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 "KROF", or equivalently row n is obtained as KROF(SPINE(n-th row of A089840)). See A122202 and A122203 for the description of KROF and SPINE. Moreover, each row of A122288 can be obtained as the "NEPEED" transform of the corresponding row in A122285. (See A122284 for the description of NEPEED). Each row occurs only once in this table. Inverses of these permutations can be found in table A122287. This table contains also all the rows of A122202 and A089840.

This permutation is induced by a wreath recursion a = s(a,b), b = (b,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 103 of the Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end, continuing complementing as long as the first 1-bit is reached, which is the last bit to be complemented.

The automorphism group of infinite binary tree (isomorphic to an infinitely iterated wreath product of cyclic groups of two elements) embeds naturally into the group of "size-preserving Catalan bijections". Scheme-function psi gives an isomorphism that maps this kind of permutation to the corresponding Catalan automorphism/bijection (that acts on S-expressions). The following identities hold: *A069770 = psi(A063946) (just swap the left and right subtrees of the root), *A057163 = psi(A054429) (reflect the whole tree), *A069767 = psi(A153141), *A069768 = psi(A153142), *A122353 = psi(A006068), *A122354 = psi(A003188), *A122301 = psi(A154435), *A122302 = psi(A154436) and from *A154449 = psi(A154439) up to *A154458 = psi(A154448). See also comments at A153246 and A153830.

a(1) to a(2^n) is the sequence of row sequency numbers in a Hadamard-Walsh matrix of order 2^n, when constructed to give "dyadic" or Payley sequency ordering. - Ross Drewe, Mar 15 2014

In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the denominator into the numerator: A007305(n) = A047679(a(n)). - Yosu Yurramendi, Aug 01 2020

This subset of integers is closed by the actions of A069770, A057163, A069767, A069768, A122353, A122354, A122301, A122302, etc. (meaning, e.g., that A069767(a(n)) is a member from this sequence for all n), that is, by any Catalan bijection which is an image of some element of the automorphism group of infinite binary tree (the latter in a sense given by Grigorchuk, et al., being isomorphic to an infinitely iterated wreath product of cyclic groups of two elements). See the comments about the isomorphism "psi" given at A153141.

a(n) could be probably computed directly from the binary expansion of n by using a (somewhat) similar ranking function as given in A209640, but utilizing A009766 instead of A007318.

The signature-permutation of the automorphism which is derived from the fifteenth non-recursive automorphism *A089859 with recursion schema FORK (see A122201 for the definition).