We introduce a new symbolic representation based on an original generalization of counter abstraction. Unlike classical counter abstraction (used in the analysis of parameterized systems with unordered or unstructured topologies) the new representation is tailored for proving properties of linearly ordered parameterized systems, i.e., systems with arbitrary many finite processes placed in an array. The relative positions in the array capture the relative priorities of the processes. Configurations of such systems are finite words of arbitrary lengths. The processes communicate using global transitions constrained by their relative priorities. Intuitively, an element of the symbolic representation has a base and a set of counters. It denotes configurations that respect the constraints imposed by the counters and that have the base as a sub word. We use the new representation in a uniform and automatic Counter Example Guided Refinement scheme. We introduce a relaxation operator that allows a well quasi ordering argument for the termination of each iteration of the refinement loop. We explain how to refine the relaxation to systematically prune out false positives. We implemented a tool to illustrate the approach on a number of parameterized systems.