Optimal word chains for the Thue-Morse word

Optimal Word Chains for the Thue-Morse Word

Word chains are an extension of addition chains to words and can be used as a complexity measure for languages. Let Z= {a, b} and cp be the morphism cp: Z* +.?J* given by cp(a)=ab and cp(b)= ba. We study word chains for the iterates $‘(a) of the Thue-Morse word. The length of optimal word chains for #‘(a) is proved to be 2n 1, and a conjecture on the enumeration of optimal word chains computing...

On the 2-abelian complexity of the Thue-Morse word

We show that the 2-abelian complexity of the infinite Thue–Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue–Morse word.

On the powers in the Thue-Morse word

Anti-Power Prefixes of the Thue-Morse Word

Recently, Fici, Restivo, Silva, and Zamboni defined a k-anti-power to be a word of the form w1w2 · · ·wk, where w1, w2, . . . , wk are distinct words of the same length. They defined AP (x, k) to be the set of all positive integers m such that the prefix of length km of the word x is a k-anti-power. Let t denote the Thue-Morse word, and let F(k) = AP (t, k) ∩ (2Z+ − 1). For k > 3, γ(k) = min(F(...

Sierpinski’s Triangle and the Prouhet-Thue-Morse Word

Sierpinski's triangle is a fractal and the Prouhet-Thue-Morse word is suÆciently chaotic to avoid cubes. Here we observe that there is at least a tenuous connection between them: the Sierpinski triangle is evident in Pascal's triangle mod 2 whose inverse, as an in nite lower-triangular matrix, involves the Prouhet-Thue-Morse word. Pascal's triangle mod 2 (Fig. 1b) is a discrete version of the f...