Characterizations of families of morphisms and words via binomial complexities

Rigo, Michel; Stipulanti, Manon; Whiteland, Markus

doi:10.1016/j.ejc.2024.103932

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Characterizations of families of morphisms and words via binomial complexities

Rigo, Michel; Stipulanti, Manon; Whiteland, Markus

2024 • In European Journal of Combinatorics

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https://hdl.handle.net/2268/308062

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10.1016/j.ejc.2024.103932

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Keywords :

Factor complexity; Abelian complexity; Binomial complexity; powers of the Thue– Morse morphism; Sturmian words; Combinatorics on words

Abstract :

[en] Two words are k-binomially equivalent if each subword of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a counting function that maps n to the number of k-binomial equivalence classes represented by its factors of length n. Cassaigne et al. [Int. J. Found. Comput. S., 22(4) (2011)] characterized a family of morphisms, which we call Parikh-collinear, as those morphisms that map all words to words with bounded 1-binomial complexity. Firstly, we extend this characterization: they map words with bounded k-binomial complexity to words with bounded (k + 1)-binomial complexity. As a consequence, fixed points of Parikh-collinear morphisms are shown to have bounded k- binomial complexity for all k. Secondly, we give a new characterization of Sturmian words with respect to their k-binomial complexity. Then we characterize recurrent words having, for some k, the same j-binomial complexity as the Thue–Morse word for all j ≤ k. Finally, inspired by questions raised by Lejeune, we study the relationships between the k- and (k + 1)-binomial complexities of infinite words; as well as the link with the usual factor complexity.

Disciplines :

Mathematics

Author, co-author :

Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes

Stipulanti, Manon ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes

Whiteland, Markus ; Université de Liège - ULiège > Mathematics

Language :

English

Title :

Characterizations of families of morphisms and words via binomial complexities

Publication date :

2024

Journal title :

European Journal of Combinatorics

ISSN :

0195-6698

eISSN :

1095-9971

Publisher :

Elsevier, Atlanta, Georgia

Peer reviewed :

Peer Reviewed verified by ORBi

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]

Available on ORBi :

since 23 October 2023

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