This paper explores optimal control problems with l(1) objectives involving linear discrete-time systems. These problems can be efficiently solved as linear programs. They also have previously been shown to yield sparse solutions, including idle or deadbeat solutions where the input or output is respectively zero along the entire control horizon. The main contribution of this paper is to derive conditions on the problem parameters that specify when idle or deadbeat solutions occur. These results, based on analyzing the dual problem, demonstrate how different types of sparse solutions result from the choice of the problem parameters and, as a consequence, may guide the design of controllers employing l(1) objectives.