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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.14234 |
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| _version_ | 1866910218799022080 |
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| author | MacAulay, Noah |
| author_facet | MacAulay, Noah |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14234 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Group Theory of the Kolakoski Sequence MacAulay, Noah Group Theory 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 $n$. |
| title | Group Theory of the Kolakoski Sequence |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.14234 |