Linear allocation functions are commonly used in mapping programs expressed as systems of recurrence equations to systolic arrays. The interconnections in a systolic array are usually required to belong to a small set of permissible vectors. Thus, the space of all systolic arrays that can be derived from a given system of recurrences (program) is limited, regardless of the program being mapped. By in­vestigating the nature of this constraint, we derive upper bounds on the number of possible systolic arrays that can be derived. These bounds are surprisingly small: there can be no more than 4 linear systolic implementations of 2-dimensional re­currences, and no more than 13 (purely systolic) planar arrays for a 3-dimensional system of recurrences. We present an efficient procedure to utilize these bounds to generate all possible linear allocation functions for a given system of recurrences, and show how it may be used for the computer-aided design of optimal systolic arrays.