Reflection on the Reflection Complexity

Abstract The factor complexity $$\mathcal {C}_{{\textbf {u}}}$$ C u of a sequence $$\textbf{u}= \varvec{u}_{0}\varvec{u}_{1}\varvec{u}_{2} \cdots $$ u = u 0 u 1 u 2 ⋯ over a finite alphabet counts the number of factors of length n occurring in $$\textbf{u}$$ u , i.e., $$\mathcal {C}_\textbf{u}{(n)} = \#{\mathcal {L}}_n(\textbf{u})$$ C u ( n ) = # L n ( u ) , where $${\mathcal {L}}_n\varvec{(}\textbf{u}\varvec{)}= {\{}\varvec{u}_{{i}}\varvec{u}_{{i+1}}\cdots \varvec{u}_{{i+n-1}}{: i} \in \mathbb {N}{\}}$$ L n ( u ) = { u i u i + 1 ⋯ u i + n - 1 : i ∈ N } . Two factors of $${\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}$$ 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 $${r}_\textbf{u}$$ r u which counts the number of non-equivalent factors of $${\mathcal {L}}_{{n}}\varvec{(}\textbf{u}\varvec{)}$$ L n ( u ) . They formulated the following conjecture: a sequence $$\textbf{u}$$ u