Combinatorics on finite words and the length of a finite-dimensional associative algebra

Let $f_W(n)$ be the number of different factors of length $n$ appearing in $W$. A classical result of Morse and Hedlund, stated in 1938, asserts that an infinite word $W$ is ultimately periodic if and only if $f_W(n)\le n$ for some $n\in \mathbb{N}$. In this paper, we describe the form of finite words that satisfy the condition $f_W(n)\le n$. We study relations between power avoidance and subword complexity of a finite word. We apply our combinatorial results to study the interrelations between various numerical invariants of finite-dimensional associative algebras.