cp's OEIS Frontend

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 "ENIPS", or equivalently row n is obtained as ENIPS(SPINE(n-th row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122286. This table contains also all the rows of A122204 and A089840.

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153142. See further comments there and in A153141.

This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(*A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack.

Note: the name in Finnish is "Naks".

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153141. See further comments there.

This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.

Note: the name in Finnish is "Niks".

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "SPINE", or equivalently row n is obtained as SPINE(ENIPS(n-th row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122285. This table contains also all the rows of A122203 and A089840.

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.

A005517 and A005518 give the minimum and maximum value occurring in each such range.

Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).

If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.

A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

This is a permutation of natural numbers induced when Euler's triangulation of convex polygons, encoded by the sequence A014486 in a straightforward way (via binary trees, cf. the illustration of the rotation of a triangulated pentagon, given in the Links section) are rotated clockwise.

In A057161 and A057162, the cycles between A014138(n-1)-th and A014138(n)-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A001683(n+2) equivalence classes of flexagons (or unlabeled plane boron trees), thus the latter sequence can be counted with the Maple procedure A057162_CycleCounts given below. Cf. also the comments in A057161.

This automorphism reflects the interpretations (pp)-(rr) of Stanley, obtained from the Dyck paths with the "rising slope mapping" illustrated on the example lines.

Deutsch and Elizalde show in their paper that this automorphism converts certain properties concerning "tunnels" of Dyck path to another set of properties concerning the number of hills, even and odd rises, as well as the number of returns (A057515), thus proving the equidistribution of the said parameters.

This automorphism is implemented with function "tau" (Scheme code given below) that takes as its arguments an S-expression and a Catalan automorphism that permutes only the top level of the list (i.e., the top-level branches of a general tree, or the whole arches of a Dyck path) and thus when the permuting automorphism is applied to a list (parenthesization) of length 2n it induces some permutation of [1..2n].

This automorphism is induced in that manner by the automorphism *A127287 and likewise, *A127289 is induced by *A127285, *A057164 by *A057508, *A057501 by *A057509 and *A057502 by *A057510.

Note that so far these examples seem to satisfy the homomorphism condition, e.g., as *A127287 = *A127285 o *A057508 so is *A127291 = *A127289 o *A057164. and likewise, as *A057510 = *A057508 o *A057509 o *A057508, so is *A057502 = *A057164 o *A057501 o *A057164.

However, it remains open what are the exact criteria of the "picking automorphism" and the corresponding permutation that this method would induce a bijection. For example, if we give *A127288 (the inverse of *A127287) to function "tau" it will not induce *A127292 and actually not a bijection at all.

Instead, we have to compute the inverse of this automorphism with another, more specific algorithm that implements Deutsch's and Elizalde's description and is given in A127300.