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 Facw(n) be the set of factors of length n of the word w, and let Palw(n) ⊆ Facw(n) be the set of palindromic factors of length n of the word w. We present several upper bounds for |Facw(n)| and |Palw(n)|, where w is a rich word. Let δ = [see formula in PDF]. In particular we show that |Facw(n)| ≤ (4q2n)δ ln 2n+2. In 2007, Baláži, Masáková, and Pelantová showed that |Palw(n)|+|Palw(n+1)| ≤ |Facw(n+1)|-|Facw(n)|+2, where w is an infinite word whose set of factors is closed under reversal. We prove this inequality for every finite word v with |v| ≥ n + 1 and v(n + 1) closed under reversal.