Abstract
Inspired by questions raised by Lejeune, we study the relationships between the k and \((k+1)\)-binomial complexities of an infinite word; as well as the link with the usual factor complexity. We show that pure morphic words obtained by iterating a Parikh-collinear morphism, i.e., a morphism mapping all words to words with bounded abelian complexity, have bounded k-binomial complexity. We further study binomial properties of the images of aperiodic binary words in general, and Sturmian words in particular, by a power of the Thue–Morse morphism.
M. Stipulanti and M. Whiteland—Supported by the FNRS Research grants 1.B.397.20F and 1.B.466.21F respectively. M. Whiteland dedicates this paper to the memory of his father Alan Whiteland (1940–2021).
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Notes
- 1.
We define \(\prec \) deliberately with “infinitely many” rather than “all large enough”: for the period-doubling word \(\mathbf {pd}\) (fixed point of \(0\mapsto 01\), \(1\mapsto 00\)) there exist infinitely many n and m such that \(\mathsf {b}_{\mathbf {pd}}^{(2)}(n) = \mathsf {p}_{\mathbf {pd}}(n)\) and \(\mathsf {b}_{\mathbf {pd}}^{(2)}(m) < \mathsf {p}_{\mathbf {pd}}(m)\) [9, Prop. 4.5.1].
- 2.
We warn the reader that the term \(\varphi \)-factorization has a different meaning in [10]. Our \(\varphi ^j\)-factorization corresponds to their “factorization of order j”.
- 3.
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Rigo, M., Stipulanti, M., Whiteland, M.A. (2022). Binomial Complexities and Parikh-Collinear Morphisms. In: Diekert, V., Volkov, M. (eds) Developments in Language Theory. DLT 2022. Lecture Notes in Computer Science, vol 13257. Springer, Cham. https://doi.org/10.1007/978-3-031-05578-2_20
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