Moments of conjugacy classes of binary words

AbstractFor each nonempty binary word w=c1c2⋯cq, where ci∈{0,1}, the nonnegative integer ∑i=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, (e) N(w) is a set of |w| consecutive integers and 0∈N(w), (f) w is primitive and [w]⊂St