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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.

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.

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 permutation is induced by a wreath recursion a = s(a,b), b = (b,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 103 of the Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end, continuing complementing as long as the first 1-bit is reached, which is the last bit to be complemented.

The automorphism group of infinite binary tree (isomorphic to an infinitely iterated wreath product of cyclic groups of two elements) embeds naturally into the group of "size-preserving Catalan bijections". Scheme-function psi gives an isomorphism that maps this kind of permutation to the corresponding Catalan automorphism/bijection (that acts on S-expressions). The following identities hold: *A069770 = psi(A063946) (just swap the left and right subtrees of the root), *A057163 = psi(A054429) (reflect the whole tree), *A069767 = psi(A153141), *A069768 = psi(A153142), *A122353 = psi(A006068), *A122354 = psi(A003188), *A122301 = psi(A154435), *A122302 = psi(A154436) and from *A154449 = psi(A154439) up to *A154458 = psi(A154448). See also comments at A153246 and A153830.

a(1) to a(2^n) is the sequence of row sequency numbers in a Hadamard-Walsh matrix of order 2^n, when constructed to give "dyadic" or Payley sequency ordering. - Ross Drewe, Mar 15 2014

In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the denominator into the numerator: A007305(n) = A047679(a(n)). - Yosu Yurramendi, Aug 01 2020

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 A057163-conjugate of the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", i.e. row n is obtained as ENIPS(A057163 o SPINE(A089840[n]) o A057163). 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 A130403. This table contains also all the rows of A122204 and A089840.

This sequence can be also obtained by starting complementing n's binary expansion from the second most significant bit, continuing towards lsb-end until the first 0-bit is reached, which is the last bit to be complemented.

In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the numerator into the denominator: A047679(n) = A007305(a(n)). - Yosu Yurramendi, Aug 30 2020

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.

If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.

A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - 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.