Palindromic factorization of rich words

TL;DR

This paper improves bounds on the length of UPS-factorization in rich words, showing it grows slower than previously known, with the new bound involving an exponential of a square root of the logarithm of word length.

Contribution

The authors establish a tighter upper bound on the UPS-factorization length for rich words, refining earlier results by introducing a bound involving exponential decay with respect to the square root of the logarithm.

Findings

Improved the upper bound on UPS-factorization length from linear to sublinear growth.

Demonstrated the bound depends on the alphabet size.

Provided explicit constants for the new bound.

Abstract

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w_1, w_2,\dots,w_p$ such that $w=w_pw_{p-1}\cdots w_1$ and $w_i$ is the longest palindromic suffix of $w_pw_{p-1}\cdots w_i$, where $1\leq i\leq p$. This palindromic factorization is called \emph{UPS-factorization}. Let $luf(w)=p$ be \emph{the length of UPS-factorization} of $w$. In 2017, it was proved that there is a constant $c$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then $luf(w)\leq c\frac{n}{\ln{n}}$. We improve this result as follows: There are constants $\mu, \pi$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then \[luf(w)\leq \mu\frac{n}{e^{\pi\sqrt{\ln{n}}}}\mbox{.}\] The constants $c,\mu,\pi$ depend on the size of the alphabet.