Palindromic factorization of rich words

Josef Rukavicka
Czech Technical University in Prague

Date(s) : 05/12/2022   iCal
15 h 00 min - 16 h 00 min

A finite word $w$ is called //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 //UPS-factorization//. Let $luf(w)=p$ be 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.


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