Moments of conjugacy classes of binary words

Abstract

For each nonempty binary word w= c 1 c 2⋯ c 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) : 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.