cp's OEIS Frontend

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.

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

This is a "multiplication table" of an infinite enumerable group. Each row and column is a permutation of A001477.

Each row of this table gives the counts of elements fixed by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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...............C...B

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

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

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

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

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

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

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...............A...C

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

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

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

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

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

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

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...........C...B

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

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

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

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

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

In terms of S-expressions, this automorphism swaps caar and cdr of an S-exp if possible, i.e., if car-side is not ().

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

The signature-permutation of the Catalan automorphism which is derived from the fourteenth non-recursive Catalan automorphism *A073269 with recursion schema SPINE (see A122203 for the definition).

The signature-permutation of the Catalan automorphism which is derived from the fourteenth non-recursive Catalan automorphism *A073269 with recursion schema ENIPS (see A122204 for the definition).