Computing the k-binomial complexity of generalized Thue--Morse words
| Autor: | Golafshan, M., Rigo, M., Whiteland, M. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 L\"u, 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. Comment: 40 pages, 9 figures, submitted for publication |
| Databáze: | arXiv |
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