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We study the number of $0,1$-words where the fraction of 0 is "almost" fixedfor any initial subword. It turns out that this study use and reveal thestructure of the Galois group (the monodromy group) of the polynomials$(x+1)^n-\lambda x^p$. ($p$ is not necessary a prime here.)

Balanced $0,1$-words and the Galois group of $(x+1)^n-\lambda x^p$