Polycubes with Small Perimeter Defect

In this paper, we consider enumeration of d-dimensional polycubes, whose perimeter (defined as the number of empty cells neighboring the polycube) has a fixed deviation from the maximum possible value. We provide a general framework for deriving such formulae, as well as several explicit formulae. In particular, we prove that for any fixed dimension d, the generating function that enumerates polycubes with a fixed defect (with respect to their volume) is rational. Moreover, its denominator is a product of cyclotomic polynomials.