On minimal factorizations of words as products of palindromes

TL;DR

This paper investigates the palindromic length of factors in infinite words, showing that under certain conditions, factors can have arbitrarily large palindromic length, addressing an open question in combinatorics on words.

Contribution

It provides a partial answer to whether infinite words can have all factors with bounded palindromic length, proving that under the (k,l)-condition, this is not possible.

Findings

Factors in words satisfying the (k,l)-condition can have arbitrarily large palindromic length.

The result applies to all k-power-free words and the Sierpinski word.

Bounded palindromic length does not hold for these classes of words.

Abstract

Given a finite word u, we define its palindromic length |u|_{pal} to be the least number n such that u=v_1v_2... v_n with each v_i 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? We give a partial answer to this question by proving that if an infinite word w satisfies the so-called (k,l)-condition for some k and l, then for each positive integer P there exists a factor u of w whose palindromic length |u|_{pal}>P. In particular, the result holds for all the k-power-free words and for the Sierpinski word.