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- W2904113304 abstract "Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $Psi^ell(w)$ for an arbitrary morphism $Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation." @default.
- W2904113304 created "2018-12-22" @default.
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- W2904113304 date "2018-12-18" @default.
- W2904113304 modified "2023-09-27" @default.
- W2904113304 title "Computing the $k$-binomial complexity of the Thue--Morse word" @default.
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