Upper Bound for Palindromic and Factor Complexity of Rich Words

TL;DR

This paper establishes new upper bounds on the number of factors and palindromic factors in rich words, extending known inequalities and analyzing their combinatorial properties.

Contribution

It provides several new upper bounds for factor and palindromic factor counts in rich words, generalizing previous inequalities to finite words.

Findings

Derived an upper bound for total factors in rich words: (q+1)8n^2(8q^{10}n)^{log_2(2n)}+q.

Extended Balázsi, Masáková, and Pelantová's inequality to finite words.

Provided insights into the combinatorial structure of rich words.

Abstract

A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word (finite or infinite) over an alphabet with $q>1$ letters, let $F(w,n)$ be the set of factors of length $n$ of the word $w$, and let $F_p(w,n)\subseteq F(w,n)$ be the set of palindromic factors of length $n$ of the word $w$. We present several upper bounds for $| F(w,n)|$ and $| F_p(w,n)|$, where $w$ is a rich word. In particular we show that \[| F(w,n)| \leq (q+1)8n^2(8q^{10}n)^{\log_2{2n}}+q\mbox{.}\] In 2007, Bal{\'a}{\v z}i, Mas{\'a}kov{\'a}, and Pelantov{\'a} showed that \[| F_p(w,n)| +| F_p(w,n+1)| \leq | F(w,n+1)|-| F(w,n)|+2\mbox{,}\] where $w$ is an infinite word whose set of factors is closed under reversal. We generalize this inequality for finite words.