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This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123496 with the recursion schema FORK (defined in A122201). - Antti Karttunen, Oct 11 2006

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

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 "FORK". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the both branches (new ones, possibly changed by the given automorphism). I.e. this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures FORK and !FORK can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122202.

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

This sequence gives two alternative ways to encode rooted plane binary trees (Stanley's interpretation 'c' = interpretation 'd' without the outermost edges):

A: scan each term from left to right and for each 0, add a leaf node to the tree (terminate a branch), for each 1, add a leftward leaning branch \, for each 2, add a rightward leaning branch / and for each 3, add a double-branch \/ and continue in left-to-right, depth-first fashion.

B: Like method A, but the roles of digits 1 and 2 are swapped. When one compares the generated trees to the "standard order" as specified in the illustrations for A014486, one obtains the permutation A074684/A074683 for the case A and A082356/A082355 for the case B.

If we assign the following weights for each digit: w(0) = -1, w(1) = w(2) = 0, w(3) = +1, then the sequence gives all base-4 numbers for which all the partial sums of digit weights (from the most significant to the least significant end) are nonnegative and the final sum is zero. The initial term 0 is considered to have no significant digits at all, so its total weight is zero also.