cp's OEIS Frontend

Shifted once right (beginning as 1,1,1,2,5,9,...) this is maximum cycle size (in the same range) of permutations A085169/A085170, shifted twice right (beginning as 1,1,1,1,2,5,9,...) this is the maximum cycle size in permutations A089867/A089868 and A089869/A089870.

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 "KROF". In this recursion scheme the algorithm first recurses down to the both branches, before the given automorphism is applied at the root of binary tree. I.e., this corresponds to the post-order (postfix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures KROF and !KROF 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 A122201.

The recursion scheme KROF is equivalent to a composition of recursion schemes ENIPS (described in A122204) and NEPEED (described in A122284), i.e., KROF(f) = NEPEED(ENIPS(f)) holds for all Catalan automorphisms f. Because of the "universal property of folds", these recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Specifically, if g = KROF(f), then (f s) = (g (cons (g^{-1} (car s)) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme KROF, we compose g with its own inverse applied to the car- and cdr-branches of a S-expression (i.e., the left and right subtrees in the context of binary trees). This implies that for any nonrecursive automorphism f of the table A089840, KROF^{-1}(f) is also in A089840, which in turn implies that all rows of table A089840 can be found also in table A122202 (e.g., row 1 of A089840 (A069770) occurs here as row 1654720) and furthermore, the table A122290 contains the rows of both tables, A122202 and A089840 as its subsets. Similar notes apply to recursion scheme FORK described in A122201. - Antti Karttunen, May 25 2007

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122203 with the recursion scheme "KROF", or equivalently row n is obtained as KROF(SPINE(n-th row of A089840)). See A122202 and A122203 for the description of KROF and SPINE. Moreover, each row of A122288 can be obtained as the "NEPEED" transform of the corresponding row in A122285. (See A122284 for the description of NEPEED). Each row occurs only once in this table. Inverses of these permutations can be found in table A122287. This table contains also all the rows of A122202 and A089840.

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

A...B..............B...C

.\./................\./

..x...C..-->.....A...x................()..B.......B..()

...\./............\./..................\./...-->...\./.

....x..............x....................x...........x..

((a . b) . c) -> (a . (b . c)) __ (() . b) --> (b . ())

That is, we rotate the binary tree right, in case it is possible and otherwise (if the left hand side of a tree is a terminal node) swap the right and left subtree (so that the terminal node ends to the right hand side), i.e. apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

See also the comments at A074679.

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.)