Prefix palindromic length of the Thue-Morse word
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The prefix palindromic length PPL_u(n) of an infinite word u is the minimal number of palindromes to which the prefix of length n of u can be decomposed. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that PPL_u(n) is unbounded for every infinite word u which is not ultimately periodic. Up to now, the conjecture has been proved only for some particular cases, including all words avoiding some power k. However, even in that case the existing upper bound for the minimal number n such that PPL_u(n)>K is greater than any constant to the power K. Precise values of PPL_u(n) are not known even for simplest examples like the Fibonacci word. In this paper, we give a first example of such a precise computation and compute the function of the prefix palindromic length of the Thue-Morse word, a famous test object for all functions on infinite words. It happens that the sequence (PPL_t(n)) is 2-regular, which raises the question if it is the case for all automatic sequences.
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