Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems

Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on deriving recurrences. We show how to use the guidelines strategically to solve some nontrivial enumeration problems in the theory of partitions and compositions. This includes a strikingly different approach to lecture hall-type theorems, with new $q$-series identities arising in the process. For completeness, we prove that the guidelines suffice to find the generating function for any system of homogeneous linear inequalities with integer coefficients. The guidelines can be viewed as a simplification of MacMahon's partition analysis with ideas from matrix techiniques, Elliott reduction, and ``adding a slice''.