Prefix palindromic length of the Thue-Morse word

Abstract

The prefix palindromic length $PPL_u(n)$ of an infinite word $u$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $u$. 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 proven for almost all words, including all words avoiding some power $p$. However, even in that simple 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 the 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 this sequence is $2$-regular, which raises the question if this fact can be generalized to all automatic sequences.