Moments of conjugacy classes of binary words

For each nonempty binary word w=c₁c₂c_q, where c_i{0,1}, the nonnegative integer ∑_i=1^q (q+1?i)c_i 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): uw]}, N(w)={M(u)?M(w): uw]} and δ(w)=max{M(u)?M(v): u,vw]}. Using these objects, we obtain equivalent conditions for a binary word to be an -word (respectively, a power of an -word). For instance, we prove that the following statements are equivalent for any binary word w with |w|2: (a) w is an -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 0N(w), (f) w is primitive and w]St.