Reflection on the Reflection Complexity

The factor complexity 𝒞_u of a sequence u= u_0u_1u_2⋯ over a finite alphabet counts the number of factors of length n occurring in u , i.e., 𝒞_u(n) = #ℒ_n(u) , where ℒ_n(u)= {u_iu_i+1⋯u_i+n-1: i∈ℕ} . Two factors of ℒ_n(u) are said to be equivalent if they are equal or one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity r_u which counts the number of non-equivalent factors of ℒ_n(u) . They formulated the following conjecture: a sequence u is eventually periodic if and only if r_u(n+2) = r_u(n) for some n∈ℕ . Here we prove the conjecture and characterize the sequences for which r_u(n+2) = r_u(n)+1 for every n∈ℕ and also the sequences for which the equality is satisfied for every sufficiently large n∈ℕ .