cp's OEIS Frontend

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

Recursive transformation SPINE for Catalan bijections has a well-defined inverse (see the definition & comments at A122203). For all Catalan bijections in A089840 that inverse produces a bijection which is itself in A089840. This sequence gives the indices to those positions where each ("primitive", non-recursive bijection) of A089840(n) occurs "atavistically" amongst the more complex recursive bijections in A122203. I.e. A122203(a(n)) = A089840(n). Similarly, other "atavistic forms" resurface as: A122288(a(n)) = A122202(n), A122285(a(n)) = A122204(n) and A122201(a(n)) = A122283(n). See also comments at A153832.

Other known terms: a(17)-a(44): 65352, 65359, 65604, 65739, 251, 1656303, 1656426, 1656552, 1656628, 1656479, 1661655, 1661816, 1666720, 1684006, 1684221, 1667042, 1667007, 1684152, 1661799, 1661676, 1666759, 1684081, 1684437, 1667151, 1684509, 1667187, 1661961, 1661944.

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.

These elements form a subgroup in A089840 (A089839).

The terms give the positions to bijections in A089840 which preserve A153835/A127302 (the non-oriented form of binary trees), but do not extend uniquely to automorphisms of an infinite binary tree.

The terms give the positions of bijections in A089840 which preserve A129593, but not A127301.

The comments at A123694 concerning counts of fixed points apply also here.