cp's OEIS Frontend

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.

This is a rare example of Catalan Automorphism (with simple definition) where the cycle count sequence (A089411) is not monotone. (See A127296 for more complex example.)

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.

This is a rare example of a simply defined Catalan Automorphism where the cycle count sequence (A089411) is not monotone. (See A127296 for a much more complex example.)

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Deutsch shows in his 2000 paper that this automorphism converts any ordered tree with the number of nodes having degree q to a tree with an equal number of odd-level nodes having degree q-1, from which it follows that, for each positive integer q, the parameters "number of nodes of degree q" and "number of odd-level nodes of degree q-1" are equidistributed. He also shows that this automorphism converts any tree with k leaves to a tree with k even-level nodes, i.e., in OEIS terms, A057514(n) = A126305(A125981(n)).

To obtain the signature permutation, we apply the given Scheme-function *A125981 to the parenthesizations as encoded and ordered by A014486/A063171 (and surrounded by extra pair of parentheses to make a valid Scheme-list) and for each n, we record the position of the resulting parenthesization (after discarding the outermost parentheses from the Scheme-list) in A014486/A063171 and this value will be a(n).

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Is the non-monotone notch from a(4)=7 to a(5)=6 the only one?