Uniformly Scattered Factors

A word u appears as a factor of another word v as it is: in one piece. When u is a subword of v, u may be scattered as several factors. We consider the case in between and put some restrictions on the number of factors as to which u is allowed to be scattered. A large class of partial orders which are generalizations of factors and subwords is obtained. Investigating the borderline between their finite and infinite antichains, we are able to fully characterize the property of being well partial order. The result generalizes Higman’s theorem.