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

TL;DR

This paper explores the combinatorial properties of finite words, characterizes those with limited subword complexity, and applies these findings to analyze numerical invariants of finite-dimensional associative algebras.

Contribution

It provides a detailed description of finite words with subword complexity at most n and connects these combinatorial properties to algebraic invariants.

Findings

Finite words with $f_W(n) \,\leq\, n$ are characterized.

Relations between power avoidance and subword complexity are established.

Applications to numerical invariants of associative algebras are demonstrated.

Abstract

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 $f_W(n)\leq n$ for some $n\in \mathbb N$. In this paper, we describe the form of finite words that satisfy the condition $f_W(n)\leq n$. 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.