Upper Bound for Palindromic and Factor Complexity of Rich Words

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.