Complexity of cutting words on regular tilings

We show that the complexity of a cutting word u in a regular tiling with a polyomino Q is equal to P n ( u ) = ( p + q − 1 ) n + 1 for all n ≥ 0 , where P n ( u ) counts the number of distinct factors of length n in the infinite word u and where the boundary of Q is constructed of 2 p horizontal and 2 q vertical unit segments.