On minimal words with given subword complexity

Abstract

We prove that the minimal length of a word Sn having the property that it contains exactly Fm+2 distinct subwords of length m for 1 ≤ m ≤ n is Fn + Fn+2. Here Fn is the nth Fibonacci number defined by F1 = F2 = 1 and F n = Fn-1 + Fn-2 for n > 2. We also give an algorithm that generates a minimal word Sn for each n ≥ 1.