Prefix palindromic length of the Sierpinski word

TL;DR

This paper investigates the prefix palindromic length of the Sierpinski word, providing a complete description and contributing to the understanding of its regularity properties in the context of automatic and non-automatic words.

Contribution

It introduces the prefix palindromic length of the Sierpinski word and characterizes its behavior, expanding the class of automatic words with known regularity properties.

Findings

The prefix palindromic length of the Sierpinski word is fully described.

The Sierpinski word's prefix palindromic length exhibits specific regularity properties.

This work adds to the understanding of palindromic structures in automatic sequences.

Abstract

The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to study; in particular, the conjecture that $p_{\mathbf{u}}(n)$ can be bounded only if $\mathbf{u}$ is ultimately periodic is open since 2013. A more recent conjecture concerns the prefix palindromic length of the period doubling word: it seems that it is not $2$-regular, and if it is true, this would give a rare if not unique example of a non-regular function of a $2$-automatic word. For some other $k$-automatic words, however, the prefix palindromic length is known to be $k$-regular. Here we add to the list of those words the Sierpinski word $\mathbf{s}$ and give a complete description of $p_{\mathbf{s}}(n)$.