Abstract
Let Σ be the alphabet { a, b}, and ϕ the morphism of Σ ∗ defined by ϕ( a) = ab and ϕ( b) = ba. The words ϕ n ( a) are prefixes of the famous Thue-Morse word abbabaabbaababbab…. Arnold and Brlek proved that the minimal word chains computing ϕ n ( a) have length 2 n−1. They also conjectured that the number of such chains was 19 if n⩾3. We prove here this conjecture.
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