Group Theory of the Kolakoski Sequence

TL;DR

This paper explores the group-theoretic structure of automata associated with iterated run-length decoding sequences, particularly focusing on the Kolakoski sequence, revealing their recursive and automorphism properties.

Contribution

It introduces and analyzes the transformation groups of permutation automata controlling iterated run-length decodings, connecting them to automorphism groups of binary trees and recursive group structures.

Findings

Transformation groups are subgroups of automorphism groups of binary trees.

These groups are likely equal to a recursively defined group with a weakly regular branch structure.

The number of maximal-length orbits for automata with odd n is determined.

Abstract

Run-length decoding is an operation on sequences in which a positive integer $a$ is replaced by a run(sequence of repeated elements) of length $a$. Iterated run-length decodings applied to sequences with alphabets consisting of pairs of positive integers $\{p, q\}$ have attracted attention from mathematicians, most notably in their role defining the well-known Kolakoski sequence. $n$-th-iterated run-length decodings are controlled by naturally associated permutation automata $A^{p,q}_n$. Here we study the transformation groups $\mathcal{K}^{p,q}_n$ of these automata. They are subgroups of the automorphism group of binary trees of depth $n+1$. They are naturally subgroups of(and likely equal to) a certain group $\mathcal{J}_n^{p,q}$ with an intricate recursive structure; their limit group is plausibly weakly regular branch. As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd $n$.