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 "REDRONI". In this recursion scheme the given automorphism is applied at the root of binary tree after the algorithm has recursed down the cdr-branch (the right hand side tree in the context of binary trees), but before the algorithm recurses down to the car-branch (the left hand side of the binary tree, with respect to the new orientation of branches, possibly changed by the applied automorphism). I.e. this corresponds to the reversed depth-first in-order traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures REDRONI and !REDRONI can be used to obtain such a transformed automorphism from any constructively (or respectively: destructively) implemented automorphism. Each row occurs only once in this table and similar notes as given e.g. for table A122202 apply here, e.g. the rows of A089840 all occur here as well. This transformation has many fixed points besides the trivial identity automorphism *A001477: at least *A069770, *A089859 and *A129604 stay as they are. Inverses of these permutations can be found in table A130400.

These terms form a subgroup in A089840 (A089839). Because A127301 can be computed as a fold and most of the recursive derivations of A089840 (i.e., tables A122201-A122204, A122283-A122290, A130400-A130403) are also folds, this sequence also gives the indices to those derived tables where bijections preserving A127301 occur.

These elements form a subgroup in A089840 (A089839). Such elements consists of only such clauses where each vertex stays at the same distance from the root of the binary tree and in the image tree will still be sibling to its original sibling in the pre-image tree.

Because A127302 can be computed as a fold and most of the recursive derivations of A089840 (i.e. tables A122201-A122204, A122283-A122290, A130400-A130403) are also folds, this sequence gives also the indices to those derived tables where bijections preserving A127302 occur.

These elements form a subgroup in A089840 (A089839) isomorphic to a group consisting of all finitely iterated wreath products of the form S_2 wr S_2 wr ... wr S_2 and each is an image of some finitary automorphism of an infinite binary tree. E.g. A089840(1) = *A069770 is an image of the generator A of Grigorchuk Group. See comments at A153246 and A153141.

The defining properties are propagated by all recursive transformations of A089840 which themselves do not behave differently depending whether there are internal or terminal vertices in the neighborhood of any vertex (at least the ones given in A122201-A122204, A122283-A122290, A130400-A130403), so this sequence gives also the corresponding positions in those tables.