A D0L word on an alphabet Σ =\0,1,\ldots,q-1\ is called symmetric if it is a fixed point w=\varphi(w) of a morphism \varphi:Σ ^* → Σ ^* defined by \varphi(i)=øverlinet_1 + i øverlinet_2 + i\ldots øverlinet_m + i for some word t_1t_2\ldots t_m (equal to \varphi(0)) and every i ∈ Σ ; here øverlinea means a \bmod q. We prove a result conjectured by J. Shallit: if all the symbols in \varphi(0) are distinct (i.e., if t_i ≠q t_j for i ≠q j), then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ ^* and a ∈ Σ .
Discrete Mathematics & Theoretical Computer Science
