cp's OEIS Frontend

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).