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.

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

This sequence grows so fast that it seems hopeless to count A057507 with Burnside's (orbit-counting) lemma.

Each row of this table gives the longest cycle/orbit produced 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.

For every cycle count LCM-sequence Axxxxxx in A073204 it holds that Axxxxxx(n) divides a(n). And this also applies to similar LCM-sequences induced by other "Catalan bijections", cf. A060113.

The next term (a(7)) has 184 digits. - Harvey P. Dale, Nov 21 2023