The complexity of finite smooth words over binary alphabets
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915969812660224 |
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| author | Cassaigne, Julien Henry, Raphaël |
| author_facet | Cassaigne, Julien Henry, Raphaël |
| contents | Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the most famous example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10733 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The complexity of finite smooth words over binary alphabets Cassaigne, Julien Henry, Raphaël Formal Languages and Automata Theory Combinatorics Dynamical Systems Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the most famous example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets. |
| title | The complexity of finite smooth words over binary alphabets |
| topic | Formal Languages and Automata Theory Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2603.10733 |