Computing the k-binomial complexity of generalized Thue--Morse words
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910761954050048 |
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| author | Golafshan, M. Rigo, M. Whiteland, M. |
| author_facet | Golafshan, M. Rigo, M. Whiteland, M. |
| contents | Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$ to the number of k-binomial equivalence classes represented by its factors of length n. The Thue--Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for $k\geq 3$ remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period $m^k$. We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18425 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the k-binomial complexity of generalized Thue--Morse words Golafshan, M. Rigo, M. Whiteland, M. Combinatorics Formal Languages and Automata Theory 68R15 Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$ to the number of k-binomial equivalence classes represented by its factors of length n. The Thue--Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for $k\geq 3$ remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period $m^k$. We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs. |
| title | Computing the k-binomial complexity of generalized Thue--Morse words |
| topic | Combinatorics Formal Languages and Automata Theory 68R15 |
| url | https://arxiv.org/abs/2412.18425 |