Improved Bounds on the Length of Maximal Abelian Square-free Words

A word is abelian square-free if it does not contain two adjacent subwords which are permutations of each other. Over an alphabet Σk on k letters, an abelian squarefree word is maximal if it cannot be extended to the left or right by letters from Σk and remain abelian square-free. Michael Korn proved that the length `(k) of a shortest maximal abelian square-free word satisfies 4k − 7 ≤ `(k) ≤ 6k − 10 for k ≥ 6. In this paper, we refine Korn’s methods to show that 6k−29 ≤ `(k) ≤ 6k−12 for k ≥ 8.