cp's OEIS Frontend

Used to construct A127388.

Used to construct A127380.

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.

Any n that occurs in the sequence, occurs for the first time at n (i.e. a(n)=n), thus there are no cases where a(n) > n. A001190(n+1) distinct values occur in each range [A014137(n-1)..A014138(n-1)]. This sequence is similar to A127302 in that it maps all the plane binary trees which belong to the same equivalence class of non-oriented binary trees to one and same integer and likewise, every Catalan bijection whose signature permutation SP satisfies the condition mentioned in A127302, satisfies in a similar way A153835(SP(n)) = A153835(n).

This automorphism is RIBS-transformation (explained in A122200) of the automorphism A127387.

A "fleeing tree" sequence computed for Catalan bijection CatBij gives for each binary tree A014486(n) the number of cases where, when a new V-node (a bud) is inserted into one of the A072643(n)+1 possible leaves of that tree, it follows that (CatBij tree) is not a subtree of (CatBij tree-with-bud-inserted). I.e., for each tree A014486(n), we compute Sum_{i=0}^A072643(n) (1 if catbij(n) is a subtree of catbij(A153250bi(n,i)), 0 otherwise). Here A153250 gives the bud-inserting operation. Note that for any Catalan Bijection, which is an image of "psi" isomorphism (see A153141) from the Automorphism Group of infinite binary trees, the result will be A000004, the zero-sequence. To satisfy that condition, CatBij should at least satisfy A127302(CatBij(n)) = A127302(n) for all n (clearly A057164 does not satisfy that, so we got nonzero terms here). However, that is just a necessary but not a sufficient condition. For example, A123493 & A123494 satisfy it, but they still produce nonzero sequences: A153247, A153248.