Moments of conjugacy classes of binary words

• Authors: Wai-Fong Chuan
• Venue: Theoretical Computer Science
• Year: 2004

For each nonempty binary word w = c1c2...Cq, where ci ∈ {0, 1}, the nonnegative integer Σi=1q (q + 1 - i)ci is called the moment of w and is denoted by M(w). Let [w] denote the conjugacy class of w. Define M([w]) = {M(u): u ∈ [w]}, N(w) = {M(u) - M(w): u ∈ [w]} and δ(w) = max{M(u) - M(v): u, v ∈ [w]}. Using these objects, we obtain equivalent conditions for a binary word to be an α-word (respectively, a power of an α-word). For instance, we prove that the following statements are equivalent for any binary word w with |w| ≥ 2: (a) w is an α-word, (b) δ(w)= |w| - 1, (c) w is a cyclic balanced primitive word, (d) M([w]) is a set of |w| consecutive positive integers, (c) N(w) is a set of |w| consecutive integers and 0 ∈ N(w), (f) w is primitive and [w] ⊂ St.