We develop a theoretical description for mechanically stable packings of frictional, spherical particles in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts that are found in frictional packings. This difference, or the saddle order m, represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate packings for each m, we show that the probability to obtain a packing with saddle order m at a given static friction coefficient can be expressed as a power-series expansion in the friction coefficient. We quantitatively describe the dependence of the average contact number on friction coefficient for static packings obtained from discrete element simulations for all friction coefficents in the large-system limit. This framework provides a method to calculate the structural and mechanical properties of frictional packings as a function of the assembly protocol used to create them.