cp's OEIS Frontend

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 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

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 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 21st non-recursive automorphism *A089863 with recursion schema KROF (see A122202 for the definition).