On minimal factorizations of words as products of palindromes

Given a finite word u, we define its palindromic length |u|pal to be the least number n such that u = v1v2 . . . vn with each vi a palindrome. We address the following open question: Does there exist an infinite non ultimately periodic word w and a positive integer P such that |u|pal ≤ P for each factor u of w? In this note we give a partial answer to this question. Let k be a positive integer. We prove that if an infinite word w is k-power free, then for each positive integer P there exists a factor u of w whose palindromic length |u|pal > P. We also extend this result to a wider class of words satisfying the so-called (k, l)-condition, which includes for example the Sierpinski word.