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.

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 "DEEPEN". In this recursion scheme the given automorphism is first applied at the root of general tree, before the algorithm recurses down to all subtrees. I.e., this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a general tree. The associated Scheme-procedures DEEPEN and !DEEPEN 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 A122284.

The recursion scheme FORK (described in A122201) is equivalent to a composition of recursion schemes SPINE (described in A122203) and DEEPEN, i.e., FORK(f) = DEEPEN(SPINE(f)) holds for all Catalan automorphisms f. These recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Thus we can equivalently define that DEEPEN(f) = FORK(SPINE^{-1}(f)). 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 A122283 form a (proper) subset of the rows of table A122201. E.g., row 1 of A122283 is row 21 of A122201, row 2 of A122283 is row 3613 of A122201, row 17 of A122283 is row 65352 of A122201, row 21 of A122283 is row 251 of A122201. - Antti Karttunen, May 25 2007

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 permutation is induced by the third Lamplighter group generating wreath recursion a = s(b,a), b = (a,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 104 of 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.

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 first non-recursive automorphism *A069770 with recursion schema NEPEED (see A122284 for the definition), or equivalently, derived from the fifteenth non-recursive automorphism *A089859 with recursion schema KROF (see A122202 for the definition).