cp's OEIS Frontend

Deutsch shows in his 1999 paper that this automorphism maps the number of doublerises of Dyck paths to number of valleys and height of the first peak to the number of returns, i.e., that A126306(n) = A127284(a(n)) and A126307(n) = A057515(a(n)) hold for all n.

The A000108(n-2) n-gon triangularizations can be reflected over n axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.

Composition with A057164 gives signature permutation for Donaghey's Map M (A057505/A057506). Embeds into itself in scale n:2n+1 as a(n) = A083928(a(A080298(n))). A127302(a(n)) = A127302(n) and A057123(A057163(n)) = A057164(A057123(n)) hold for all n.

Each row is a permutation of nonnegative integers induced by a Catalan bijection (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.

The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other Catalan bijection in this table, using one of the five allowed recursion types:

0 - Apply the given Catalan bijection and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of row number = 2)

1 - First recurse down to both subtrees of the old binary tree and only after that apply the given Catalan bijection. (last digit = 4)

2 - Apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)

3 - First recurse down to the right subtree of old binary tree and only after that apply the given Catalan bijection. (last digit = 8)

4 - First recurse down to the left subtree of old binary tree, after that apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree. (last digit = 0)

The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side and any Catalan bijection from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.

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 "ENIPS". In this recursion scheme the algorithm first recurses down to the right-hand side branch of the binary tree, before the given automorphism is applied at its root. This corresponds to the fold-right operation applied to the Catalan structure, interpreted e.g. as a parenthesization or a Lisp-like list, where (lambda (x y) (f (cons x y))) is the binary function given to fold, with 'f' being the given automorphism. The associated Scheme-procedures ENIPS and !ENIPS 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 A122203.

Because of the "universal property of folds", the recursion scheme ENIPS has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = ENIPS(f), then (f s) = (g (cons (car s) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme ENIPS, we compose g with its own inverse applied to the cdr-branch of a S-expression (i.e. the right subtree in the context of binary trees). This implies that for any non-recursive automorphism f in the table A089840, ENIPS^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.

This bijection effects the following transformation on the unlabeled rooted plane general trees (letters A, B, C, etc. refer to arbitrary subtrees located on those vertices):

A A A B B A A B C B A C

| --> | \ / --> \ / \ | / --> \ | /

| | \./ \./ \|/ \|/ etc.

I.e., it keeps "planted" (root degree = 1) trees intact, and swaps the two leftmost toplevel subtrees of the general trees that have a root degree > 1.

On the level of underlying binary trees that general trees map to (see, e.g., 1967 paper by N. G. De Bruijn and B. J. M. Morselt, or consider lists vs. dotted pairs in Lisp programming language), this bijection 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).

B C A C

\ / \ /

A x --> B x A () A ()

\ / \ / \ / --> \ /

x x x x

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

Note that the first clause corresponds to what is called "generator pi_0" in Thompson's group V. (See also A074679, A089851 and A154121 for other related generators.)

Look at the example section to see how this will produce the given sequence of integers.

Applying this permutation recursively down the right hand side branch of the binary trees (or equivalently, along the topmost level of the general trees) produces permutations A057509 and A057510 (that occur at the same index 2 in tables A122203 and A122204) that effect "shallow rotation" on general trees and parenthesizations. Applying it recursively down the both branches of binary trees (as in pre- or postorder traversal) produces A057511 and A057512 (that occur at the same index 2 in tables A122201 and A122201) that effect "deep rotation" on general trees and parenthesizations.

For this permutation, A127301(a(n)) = A127301(n) for all n, which in turn implies A129593(a(n)) = A129593(n) for all n, likewise for all such recursively generated bijections as A057509 - A057512. Compare also to A072797.

Also number of necklaces with 2*n beads, n white and n black (to get the correspondence, start at root, walk around outside of tree, use white if move away from the root, black if towards root).

Also number of terms in polynomial expression for permanent of generic circulant matrix of order n.

a(n) is the number of equivalence classes of n-compositions of n under cyclic rotation. (Given a necklace, split it into runs of white followed by a black bead and record the lengths of the white runs. This gives an n-composition of n.) a(n) is the number of n-multisets in Z mod n whose sum is 0. - David Callan, Nov 05 2003

a(n) is the number of cyclic equivalence classes of triangulations of a once-punctured n-gon. - Esther Banaian, May 06 2025

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.

In A057501 and A057502, the cycles between (A014138(n-1)+1)-th and (A014138(n))-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A002995(n+1) equivalence classes of planar trees, thus the latter sequence can be produced also with Maple procedure RotHandshakesPermutationCycleCounts given below.

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