Bordered Conjugates of Words over Large Alphabets

The border correlation function attaches to every word w a binary word β(w) of the same length where the ith letter tells whether the ith conjugate w′ = vu of w = uv is bordered or not. Let [u] denote the set of conjugates of the word w. We show that for a 3-letter alphabet A, the set of β-images equals β(An) = B∗ \ ([ abn−1 ] ∪D ) where D = {an} if n ∈ {5, 7, 9, 10, 14, 17}, and otherwise D = ∅. Hence the number of β-images is Bn 3 = 2 n − n−m, where m = 1 if n ∈ {5, 7, 9, 10, 14, 17} and m = 0 otherwise.