We apply an extension of the Nelson-Oppen combination method to develop a decision procedure for the non-disjoint union of theories modeling data structures with a counting operator and fragments of arithmetic. We present some data structures and some fragments of arithmetic for which the combination method is complete and effective. To achieve effectiveness, the combination method relies on particular procedures to compute sets that are representative of all the consequences over the shared theory. We show how to compute these sets by using a superposition calculus for the theories of the considered data structures and various solving and reduction techniques for the fragments of arithmetic we are interested in, including Gauss elimination, Fourier-Motzkin elimination and Groebner bases computation.