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.